Edelweiss Applied Science and Technology 
ISSN: 2576-8484 
Vol. 9, No. 7, 835-850 
2025 
Publisher: Learning Gate 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

© 2025 by the authors; licensee Learning Gate 
History: Received: 7 May 2025; Revised: 24 June 2025; Accepted: 27 June 2025; Published: 10 July 2025 
* Correspondence: mannan.iu31@uttarauniversity.edu.bd  

 
 
 
 
 

Evaluate all order of every element of higher 100, 105 and 107 order of 
group for multiplication composition 

 
Md. Abdul Mannan1*, Kanchon Kumar Bishnu2, Shoma Islam3, Md. Shafiul Alam Chowdhury4, Md. 
Amzad Hossain5, Md. Shafikul Islam6, Sahib Jada Eyakub Khan7, Hashiara Khatun8 
1Department of Mathematics, Uttara University, Dhaka, Bangladesh; mannan.iu31@gmail.com (M.A.M.). 
2Department of Computer Science, California State University, Los Angeles, California, USA; kbishnu@calstatela.edu (K.K.B.). 
3Department of Management Information Systems, International American University, Los Angeles, USA; 
shomaislam85@gmail.com (S.I.). 
4Department of Computer Science and Engineering, Uttara University, Dhaka, Bangladesh; 
shafiul.cse@uttarauniversity.edu.bd (M.S.A.C.). 
5Department of Education, Uttara University, Dhaka, Bangladesh; amzad_uuedu@uttarauniversity.edu.bd (M.A.H.). 
6Department of Computer Science and Engineering, Uttara University, Dhaka, Bangladesh; shafik.cse@gmail.com (M.S.I.). 
7Biman Bangladesh Airlines, Kurmitola, Dhaka, Bangladesh; sk2xenon@gmail.com (S.J.E.K.). 
8Department of Health Science, Panola College, Carthage, Texas, USA; hashiara7y@gmail.com (H.K.). 

 

 

Abstract: This paper aims at treating a study on the order of every element of higher 100, 105 and 107 
orders of group for multiplication composition. But the composition in G is associative; the 
multiplication composition is very significant in the order of elements of a group. We develop the order 

of a group 𝑜(𝐺), higher order of groups in different types of order and the order of elements 𝑜(𝑎) of a 

group in real numbers. Let G be a group and let 𝑎𝑛 ∈ 𝐺 be of infinite order n, then find Highest 

Common Factor (𝑖. 𝑒. (𝑛,   𝑚) 𝑑𝑒𝑛𝑜𝑡𝑒𝑠 𝐻. 𝐶. 𝐹 𝑜𝑓 𝑛 𝑎𝑛𝑑 𝑚) . The Highest Common Factor of two 
numbers is the “smallest non-zero common number” which is a multiple of both the numbers. 

So 𝑜(𝑎𝑚) =
𝑛

(𝑛,   𝑚)
  ,  where (n, m) denotes the H.C.F of n and m. If 𝑎 ∈ 𝐺 is of order n, then there exists 

an integer m for which 𝑎𝑚 = 𝑒 if m is a multiple of n, in general we use this. Then we develop orders of 
elements of a cyclic group and every element of higher order of a group. After that we find out the order 
of every element of a group for the higher orders of the group for being binary operation. 

Keywords: Multiplication Composition, Torsion Group, 𝐻. 𝐶. 𝐹. 𝑜(𝐺), 𝑜(𝑎) 

 
1. Introduction  

We propose to study the groups of order of an element of a group, order of a group, torsion group, 
mixed group subgroup, normal subgroup and the integral powers of an element of a group etc. Then we 
discuss the order of every element in the higher 100, 105 and 107 orders of group for multiplication 
composition. The group notation is o or *. We will frequently omit the symbol for the group operation 

but we will also often write the operation as · or + when it represents multiplication or addition in a 
group, and write 1or 0 for the corresponding identity elements respectively. It’s addition +, 

multiplication × or (.) is used as a binary operation. If the group operation is denoted as a multiplication, 

then an element  𝑎 ∈ 𝐺  is said to be order n if n is the least positive integer such that 𝑎𝑛 = 𝑒 or 𝑂(𝑎) ≤
𝑛 i. e., if 𝑎𝑛 = 𝑒 𝑎𝑛𝑑 𝑎𝑟 ≠ 𝑒   ∀  𝑟 ∈ 𝑁 𝑠. 𝑡.  𝑟 < 𝑛 . 

The order of a is denoted by 𝑂(𝑎). If𝑎𝑛 ≠ 𝑒   𝑓𝑜𝑟 𝑎𝑛𝑦  𝑛 ∈ 𝑁, then a is said to be of zero order or 

infinite order [1]. Let e is the identity element in (G, +). An element  𝑎 ∈ 𝐺  is said to be order n if 



836 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

+ Zn  such that 𝑛𝑎 = 𝑒 or 𝑂(𝑎) ≤ 𝑛. i. e., if 𝑛𝑎 = 𝑒 𝑎𝑛𝑑 𝑎𝑟 ≠ 𝑒   ∀  𝑟 ∈ 𝑁 𝑠. 𝑡. 0 < 𝑟 < 𝑛 .The order 

of a is denoted by 𝑂(𝑎). If 𝑛𝑎 ≠ 𝑒   𝑓𝑜𝑟 𝑎𝑛𝑦  𝑛 ∈ 𝑁, then a is said to be of zero order or infinite order 
[2]. The order of a group G and the orders of its elements give much information about the structure of 
the group. The order of any subgroup of G divides the order of G.  If H is a subgroup of G, 

then  𝑜𝑟𝑑(𝐺) / 𝑜𝑟𝑑(𝐻)  =  [𝐺 ∶  𝐻], where [𝐺 ∶  𝐻] is called the index of H in G. This is Lagrange's 

theorem; however, it is only true when G has finite order. If  𝑜𝑟𝑑(𝐺)  =  ∞, the quotient 𝑜𝑟𝑑(𝐺) /
 𝑜𝑟𝑑(𝐻) is not true. We see that the order of every element of a group divides the order of the group. 

For example, in the symmetric group shown above, where  𝑜𝑟𝑑(𝑆3)  =  6, the possible orders of the 
elements a, b, c. But there is no general formula relating the order of a product ab to the orders of a and 
b. In fact, it is possible that both a and b have finite order while ab has infinite order, or that both a and 
b have infinite order while ab has finite order. (i.e. [3-5]). But here we discuss the order of groups of 
higher odd, even and prime order of groups as 69, 70 and 71. Then we find out the order of every 
element of a group in different types of the higher even, odd and prime order of the group for 
composition [6, 7]. 
 

2.  Integral Powers of an Element of a Group 
2.1. Multiplication Composition [8, 9] 

Let (G, .) be a group. Let 𝑎 ∈ 𝐺  be an arbitrary element.  
By closure property, all the elements a, aa, aaa, .  .   .etc. belong to G. 
Since the composition in G is associative. Hence aaa  .   .   .   to n factors is independent of the 

manner in which the factors are grouped. 

If n is a positive integer, then define factorsn   to.   .   .   a a. a. =a n

 

propertyclosurebyG,a n   

If e is identity in G, then we define e.a 0 =  

If n is a negative integer, then by define ( ) ( ) n1n1nn aofinversetheisawhere,aa
−−− =  

Consequently, ( ) Ga
1n 
−

, since the inverse of every element of G belong to G.  Ga n −

 
According to the definition 

( ) ( )

( )( )( )

( )

( ) ( ) .aaa

a

factorsnto...aaa

factorsnto...aaaa

n11nn

n1

111

11n

−−−

−

−−−

−−

==

=

=

=

 
The following law of indices can be easily proved 

( )
Znm,andGaaaaand

Znm,andGaaa

nmnm

mnnm

=

=

+

 

Thus, we defined 
na for all integral values of n, positive, negative or zero. 

 

3.  Significance of the order of an element of a group 
We begin this section with the following theorem, which highlights the fundamental significance of 

the order of an element in a group. 
 



837 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

3.1. Theorem [10] 

Let G be a group and let a ∈ G be of infinite order n. Then show that O (𝑎)𝑘 = 
𝑛

(𝑛,𝑘)
 where k is any 

integer and (n, k) is denoted the highest common factor of n and k. 

Proof: Let o(a) = n ,𝑜(𝑎)𝑘 = h, (n, k) = m. 

Now we will prove that  ℎ =
𝑛

𝑚
 . 

We have,  
o(a) = n (1) 

or, 𝑎𝑛 = e or, 𝑜(𝑎)𝑘 =h  or, (𝑎𝑘)ℎ = e                                                                                                                
(2) 

or, 𝑎𝑘ℎ = e  
And, (n, k) = m 
or, n = mp , k = mg ; where p and q integers and (p, q) = 1 
From (2) we get, 

𝑎𝑘ℎ = e 

or, 𝑎𝑘ℎ = 𝑎𝑛 or, kh = n,  or, (𝑘ℎ) or, (𝑚𝑞ℎ) or, (𝑞ℎ) or, (ℎ)  where (p, q) = 1      (3) 

Now, 𝑎𝑘𝑝 = 𝑎(𝑚𝑞)𝑝 

                 = 𝑎(𝑚𝑝)𝑞[By Associative law] 

                = 𝑎𝑛𝑞 

                = (𝑎𝑛)𝑞 

                 = 𝑒𝑞 

or, 𝑎𝑘𝑝 = e (4) 

or, (𝑎𝑘)𝑝 = e 
or, o(k) = p 
or, h = p (5) 
or, h / p 
From (3) and (5) then we get, 
P = h 

or, h = 
𝑛   

𝑚
            QED 

 
3.2. Theorem [11] 

Show that the order of every element of a finite group is finite. 
Proof:  Let G be a finite group with multiplication composition. 

Let 𝑎 ∈ 𝐺 be an arbitrary element. 

Now we will prove that 𝑂(𝑎) is finite. 
By closure property, all the elements a2=a. a, a3=a. a. a,   .   .   .  etc. belong to G 
i.e. a, a2, a3, a4, a5, a6, a7,   .   .   .   etc. belong to G. 
But all these elements are not distinct. Since G is finite. 

Let e be the identity in G, then e.a 0 =  
Let us suppose that 

nmas0,nmpwheree,aea

eaaaaa

n.mwhereaa

pnm

0nnnm

nm

−===

===

=

−

−−

 
Also m and n are finite and hence p is a finite positive integer. 

Now p is a positive integer s.t. ea p = . 



838 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

This proves that  

𝑜(𝑎) ≤ 𝑝 = 𝑓𝑖𝑛𝑖𝑡𝑒𝑛𝑢𝑚𝑏𝑒𝑟 

𝑖. 𝑒. 𝑜(𝑎) ≤ 𝑎𝑓𝑖𝑛𝑖𝑡𝑒𝑛𝑢𝑚𝑏𝑒𝑟 ⇒ 𝑜(𝑎)𝑖𝑠𝑓𝑖𝑛𝑖𝑡𝑒 
 

4.  Result and Discussion 
In this section we developed the result of order of every element for multiplication composition in 

the higher 100, 105 and 107 orders of group for multiplication composition. We have used the three 
sections such that  

i. The Higher 100 Order of a Group for Multiplication Composition 
ii. The Higher 105 Order of a Group for Multiplication Composition 
iii. The Higher 107 Order of a Group for Multiplication Composition 

 
4.1.The Higher 100 Order of a Group for Multiplication Composition [12] 

Find the order of every element in the multiplication group 𝐺 = {𝑎, 𝑎2, 𝑎3, 𝑎4,   . . . , 𝑎100 = 𝑒} 
Solution: 

The identity element of the given group is 𝑎100 = 𝑒 ⇒ 𝑜(𝑎) = 100  ∴  𝑜(𝑎) = 100   

We know that 𝑜(𝑎𝑚) =
𝑛

(𝑛,   𝑚)
  ,    𝑤ℎ𝑒𝑟𝑒 (𝑛,   𝑚)𝑑𝑒𝑛𝑜𝑡𝑒𝑠 𝑡ℎ𝑒 𝐻. 𝐶. 𝐹 𝑜𝑓 𝑛 𝑎𝑛𝑑 𝑚 

To determine 𝑜(𝑎2) 

Here, (100,   2) = 𝐻. 𝐶. 𝐹 𝑜𝑓 100 𝑎𝑛𝑑 2  ∴ 𝑜(𝑎2) =
100

(100,   2)
=

100

2
= 50 ⇒ 𝑜(𝑎2) = 50  

To determine 𝑜(𝑎3) 

Here, (100,   3) = 𝐻. 𝐶. 𝐹 𝑜𝑓 100 𝑎𝑛𝑑 3  ∴ 𝑜(𝑎3) =
100

(100,   3)
=

100

1
= 100 ⇒ 𝑜(𝑎2) = 100 

To determine 𝑜(𝑎4) 

Here, (100,   4) = 𝐻. 𝐶. 𝐹 𝑜𝑓 100 𝑎𝑛𝑑 4  ∴ 𝑜(𝑎4) =
100

(100,   4)
=

100

4
= 25 ⇒ 𝑜(𝑎4) = 25  

To determine 𝑜(𝑎5) 

Here, (100,   5) = 𝐻. 𝐶. 𝐹 𝑜𝑓 100 𝑎𝑛𝑑 5  ∴ 𝑜(𝑎5) =
100

(100,   5)
=

100

5
= 20 ⇒ 𝑜(𝑎5) = 20  

To determine 𝑜(𝑎6) 

Here, (100,   6) = 𝐻. 𝐶. 𝐹 𝑜𝑓 100 𝑎𝑛𝑑 6  ∴ 𝑜(𝑎6) =
100

(100,   6)
=

100

2
= 50 ⇒ 𝑜(𝑎6) = 50  

To determine 𝑜(𝑎7) 

Here, (100,   7) = 𝐻. 𝐶. 𝐹 𝑜𝑓 100 𝑎𝑛𝑑 7  ∴ 𝑜(𝑎7) =
100

(100,   7)
=

100

1
= 100 ⇒ 𝑜(𝑎7) = 100  

To determine 𝑜(𝑎8) 

Here, (100,   8) = 𝐻. 𝐶. 𝐹 𝑜𝑓 100 𝑎𝑛𝑑 8  ∴ 𝑜(𝑎8) =
100

(100,   8)
=

100

4
= 25 ⇒ 𝑜(𝑎8) = 25  

To determine 𝑜(𝑎9) 

Here, (100,   9) = 𝐻. 𝐶. 𝐹 𝑜𝑓 100 𝑎𝑛𝑑 9  ∴ 𝑜(𝑎2) =
100

(100,   9)
=

100

1
= 100 ⇒ 𝑜(𝑎9) = 100  

To determine𝑜(𝑎10):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎10) =
100

(100,   10)
=

100

10
= 10 ⇒ 𝑜(𝑎10) = 10  

To determine 𝑜(𝑎11): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎2) =
100

(100,   11)
=

100

1
= 100 ⇒ 𝑜(𝑎11) = 100  

To determine𝑜(𝑎12):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎12) =
100

(100,   12)
=

100

4
= 25 ⇒ 𝑜(𝑎12) = 25  

To determine𝑜(𝑎13):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎13) =
100

(100,   13)
=

100

1
= 100 ⇒ 𝑜(𝑎13) = 100  

To determine𝑜(𝑎14):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎14) =
100

(100,   14)
=

100

2
= 50 ⇒ 𝑜(𝑎14) = 50  

To determine𝑜(𝑎15):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎15) =
100

(100,   15)
=

100

5
= 20 ⇒ 𝑜(𝑎15) = 20  



839 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

To determine𝑜(𝑎16):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎16) =
100

(100,   16)
=

100

4
= 25 ⇒ 𝑜(𝑎16) = 25  

To determine𝑜(𝑎17):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎17) =
100

(100,   17)
=

100

1
= 100 ⇒ 𝑜(𝑎17) = 100 

To determine𝑜(𝑎18):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎18) =
100

(100,   18)
=

100

2
= 50 ⇒ 𝑜(𝑎18) = 50  

To determine𝑜(𝑎19): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎19) =
100

(100,   19)
=

100

1
= 100 ⇒ 𝑜(𝑎19) = 100  

To determine𝑜(𝑎20):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎20) =
100

(100,   20)
=

100

20
= 5 ⇒ 𝑜(𝑎20) = 5 

To determine𝑜(𝑎21):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎21) =
100

(100,   21)
=

100

1
= 100 ⇒ 𝑜(𝑎21) = 100  

To determine𝑜(𝑎22):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎22) =
100

(100,   22)
=

100

2
= 50 ⇒ 𝑜(𝑎22) = 50  

To determine𝑜(𝑎23):𝑆𝑜 𝑡ℎ𝑎𝑡𝑜(𝑎23) =
100

(100,   23)
=

100

1
= 100 ⇒ 𝑜(𝑎23) = 100  

To determine𝑜(𝑎24): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎24) =
100

(100,   24)
=

100

4
= 25 ⇒ 𝑜(𝑎24) = 25  

To determine𝑜(𝑎25):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎25) =
100

(100,   25)
=

100

25
= 4 ⇒ 𝑜(𝑎25) = 4  

To determine𝑜(𝑎26): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎26) =
100

(100,   26)
=

100

2
= 50 ⇒ 𝑜(𝑎26) = 50  

To determine𝑜(𝑎27):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎27) =
100

(100,   27)
=

100

1
= 100 ⇒ 𝑜(𝑎27) = 100  

To determine𝑜(𝑎28):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎28) =
100

(100,   28)
=

100

4
= 25 ⇒ 𝑜(𝑎28) = 25  

To determine𝑜(𝑎29):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎29) =
100

(100,   29)
=

100

1
= 100 ⇒ 𝑜(𝑎29) = 100  

To determine𝑜(𝑎30):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎30) =
100

(100,   30)
=

100

10
= 10 ⇒ 𝑜(𝑎30) = 10  

To determine𝑜(𝑎31):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎31) =
100

(100,   31)
=

100

1
= 100 ⇒ 𝑜(𝑎31) = 100  

To determine𝑜(𝑎32):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎32) =
100

(100,   32)
=

100

4
= 25 ⇒ 𝑜(𝑎32) = 25  

To determine𝑜(𝑎33):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎33) =
100

(100,   33)
=

100

1
= 100 ⇒ 𝑜(𝑎33) = 100  

To determine𝑜(𝑎34):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎34) =
100

(100,   34)
=

100

2
= 50 ⇒ 𝑜(𝑎34) = 50 

To determine𝑜(𝑎35):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎35) =
100

(100,   35)
=

100

5
= 20 ⇒ 𝑜(𝑎35) = 20  

To determine𝑜(𝑎36): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎36) =
100

(100,   36)
=

100

4
= 25 ⇒ 𝑜(𝑎36) = 25  

To determine𝑜(𝑎37): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎37) =
100

(100,   37)
=

100

1
= 100 ⇒ 𝑜(𝑎37) = 100  

To determine𝑜(𝑎38):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎38) =
100

(100,   38)
=

100

2
= 50 ⇒ 𝑜(𝑎38) = 50  

To determine𝑜(𝑎39):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎39) =
100

(100,   39)
=

100

1
= 100 ⇒ 𝑜(𝑎39) = 100  

To determine𝑜(𝑎40):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎40) =
100

(100,   40)
=

100

10
= 10 ⇒ 𝑜(𝑎40) = 10  

To determine𝑜(𝑎41):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎41) =
100

(100,   41)
=

100

1
= 100 ⇒ 𝑜(𝑎41) = 100  

To determine𝑜(𝑎42):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎42) =
100

(100,   42)
=

100

2
= 50 ⇒ 𝑜(𝑎41) = 50  

To determine𝑜(𝑎43):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎43) =
100

(100,   43)
=

100

1
= 100 ⇒ 𝑜(𝑎43) = 100  

To determine𝑜(𝑎44):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎44) =
100

(100,   44)
=

100

4
= 25 ⇒ 𝑜(𝑎44) = 25  

To determine𝑜(𝑎45):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎45) =
100

(100,   45)
=

100

5
= 20 ⇒ 𝑜(𝑎45) = 20 



840 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

To determine𝑜(𝑎46):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎47) =
100

(100,   46)
=

100

2
= 50 ⇒ 𝑜(𝑎46) = 50 

To determine𝑜(𝑎47):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎47) =
100

(100,   47)
=

100

1
= 100 ⇒ 𝑜(𝑎47) = 100 

To determine𝑜(𝑎48):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎48) =
100

(100,   41)
=

100

4
= 25 ⇒ 𝑜(𝑎48) = 25  

To determine𝑜(𝑎49):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎49) =
100

(100,   49)
=

100

1
= 100 ⇒ 𝑜(𝑎49) = 100  

To determine𝑜(𝑎50):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎50) =
100

(100,   50)
=

100

50
= 2 ⇒ 𝑜(𝑎50) = 2  

To determine𝑜(𝑎51):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎51) =
100

(100,   51)
=

100

1
= 100 ⇒ 𝑜(𝑎51) = 100 

To determine𝑜(𝑎52):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎52) =
100

(100,   52)
=

100

2
= 50 ⇒ 𝑜(𝑎52) = 50 

To determine𝑜(𝑎53):𝑆𝑜 𝑡ℎ𝑎𝑡𝑜(𝑎53) =
100

(100,   53)
=

100

1
= 100 ⇒ 𝑜(𝑎53) = 100  

To determine𝑜(𝑎54):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎54) =
100

(100,   54)
=

100

2
= 50 ⇒ 𝑜(𝑎54) = 50  

To determine𝑜(𝑎55):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎55) =
100

(100,   55)
=

100

5
= 20 ⇒ 𝑜(𝑎55) = 20  

To determine𝑜(𝑎56):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎56) =
100

(100,   56)
=

100

2
= 50 ⇒ 𝑜(𝑎56) = 50  

To determine𝑜(𝑎57):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎57) =
100

(100,   57)
=

100

1
= 100 ⇒ 𝑜(𝑎57) = 100  

To determine𝑜(𝑎58):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎58) =
100

(100,   58)
=

100

2
= 50 ⇒ 𝑜(𝑎58) = 50  

To determine𝑜(𝑎59):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎59) =
100

(100,   59)
=

100

1
= 100 ⇒ 𝑜(𝑎59) = 100  

To determine𝑜(𝑎60):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎60) =
100

(100,   60)
=

100

20
= 5 ⇒ 𝑜(𝑎60) = 5  

To determine𝑜(𝑎61): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎61) =
100

(100,   61)
=

100

1
= 100 ⇒ 𝑜(𝑎61) = 100   

To determine𝑜(𝑎62):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎62) =
100

(100,   62)
=

100

2
= 50 ⇒ 𝑜(𝑎62) = 50  

To determine𝑜(𝑎63):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎63) =
100

(100,   63)
=

100

1
= 100 ⇒ 𝑜(𝑎63) = 100   

To determine𝑜(𝑎64):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎64) =
100

(100,   64)
=

100

4
= 25 ⇒ 𝑜(𝑎64) = 25   

To determine𝑜(𝑎65):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎65) =
100

(100,   65)
=

100

5
= 20 ⇒ 𝑜(𝑎65) = 20   

To determine𝑜(𝑎66):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎66) =
100

(100,   66)
=

100

2
= 50 ⇒ 𝑜(𝑎66) = 50   

To determine𝑜(𝑎67):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎67) =
100

(100,   67)
=

100

1
= 100 ⇒ 𝑜(𝑎67) = 100 

To determine𝑜(𝑎68):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎68) =
100

(100,   68)
=

100

2
= 50 ⇒ 𝑜(𝑎68) = 50   

To determine𝑜(𝑎69):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎69) =
100

(100,   69)
=

100

1
= 100 ⇒ 𝑜(𝑎69) = 100   

To determine𝑜(𝑎70):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎70) =
100

(100,   70)
=

100

10
= 10 ⇒ 𝑜(𝑎70) = 10  

To determine𝑜(𝑎71):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎71) =
100

(100,   71)
=

100

1
= 100 ⇒ 𝑜(𝑎71) = 100   

To determine𝑜(𝑎72):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎72) =
100

(100,   72)
=

100

4
= 25 ⇒ 𝑜(𝑎72) = 25 

To determine𝑜(𝑎73):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎73) =
100

(100,   73)
=

100

1
= 100 ⇒ 𝑜(𝑎73) = 100   

To determine𝑜(𝑎74):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎74) =
100

(100,   74)
=

100

2
= 50 ⇒ 𝑜(𝑎74) = 50   

To determine𝑜(𝑎75):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎75) =
100

(100,   75)
=

100

25
= 4 ⇒ 𝑜(𝑎75) = 4   



841 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

To determine𝑜(𝑎76):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎76) =
100

(100,   76)
=

100

4
= 25 ⇒ 𝑜(𝑎76) = 25   

To determine𝑜(𝑎77):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎77) =
100

(100,   77)
=

100

1
= 100 ⇒ 𝑜(𝑎77) = 100 

To determine𝑜(𝑎78):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎78) =
100

(100,   78)
=

100

2
= 50 ⇒ 𝑜(𝑎78) = 50   

To determine𝑜(𝑎79):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎79) =
100

(100,   79)
=

100

1
= 100 ⇒ 𝑜(𝑎79) = 100   

To determine𝑜(𝑎80):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎80) =
100

(100,   80)
=

100

20
= 5 ⇒ 𝑜(𝑎80) = 5 

To determine𝑜(𝑎81):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎81) =
100

(100,   81)
=

100

1
= 100 ⇒ 𝑜(𝑎81) = 100   

To determine𝑜(𝑎82):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎82) =
100

(100,   82)
=

100

2
= 50 ⇒ 𝑜(𝑎82) = 50 

To determine𝑜(𝑎83):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎83) =
100

(100,   83)
=

100

1
= 100 ⇒ 𝑜(𝑎83) = 100   

To determine𝑜(𝑎84):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎84) =
100

(100,   84)
=

100

4
= 25 ⇒ 𝑜(𝑎84) = 25   

To determine𝑜(𝑎85):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎85) =
100

(100,   85)
=

100

5
= 20 ⇒ 𝑜(𝑎85) = 20   

To determine𝑜(𝑎86):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎86) =
100

(100,   86)
=

100

2
= 50 ⇒ 𝑜(𝑎86) = 50   

To determine𝑜(𝑎87):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎87) =
100

(100,   87)
=

100

1
= 100 ⇒ 𝑜(𝑎87) = 100 

To determine𝑜(𝑎88):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎88) =
100

(100,   88)
=

100

4
= 25 ⇒ 𝑜(𝑎88) = 25   

To determine𝑜(𝑎89):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎89) =
100

(100,   89)
=

100

1
= 100 ⇒ 𝑜(𝑎89) = 100 

To determine𝑜(𝑎90):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎90) =
100

(100,   90)
=

100

10
= 10 ⇒ 𝑜(𝑎90) = 10 

To determine𝑜(𝑎91):𝑆𝑜 𝑡ℎ𝑎𝑡𝑜(𝑎91) =
100

(100,   91)
=

100

1
= 100 ⇒ 𝑜(𝑎91) = 100   

To determine𝑜(𝑎92):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎92) =
100

(100,   92)
=

100

2
= 50 ⇒ 𝑜(𝑎92) = 50 

To determine𝑜(𝑎93):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎93) =
100

(100,   93)
=

100

1
= 100 ⇒ 𝑜(𝑎93) = 100   

To determine𝑜(𝑎94):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎94) =
100

(100,   94)
=

100

2
= 50 ⇒ 𝑜(𝑎94) = 50   

To determine𝑜(𝑎95):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎95) =
100

(100,   95)
=

100

5
= 20 ⇒ 𝑜(𝑎95) = 20   

To determine𝑜(𝑎96):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎96) =
100

(100,   96)
=

100

2
= 50 ⇒ 𝑜(𝑎96) = 50   

To determine𝑜(𝑎97):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎97) =
100

(100,   97)
=

100

1
= 100 ⇒ 𝑜(𝑎97) = 100 

To determine𝑜(𝑎98):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎98) =
100

(100,   98)
=

100

2
= 50 ⇒ 𝑜(𝑎98) = 50   

To determine𝑜(𝑎99):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎99) =
100

(100,   99)
=

100

1
= 100 ⇒ 𝑜(𝑎99) = 100 

To determine𝑜(𝑎100):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎100) =
100

(100,   100)
=

100

100
= 1 ⇒ 𝑜(𝑎100) = 1 

 
4.2. The Higher 105 Order of a Group for Multiplication Composition [13] 

Find the order of every element in the multiplication group 𝐺 = {𝑎, 𝑎2, 𝑎3, 𝑎4,   . . . , 𝑎105 = 𝑒} 
 
Solution: 

The identity element of the given group is 𝑎105 = 𝑒 ⇒ 𝑜(𝑎) = 105  ∴  𝑜(𝑎) = 105   

We know that 𝑜(𝑎𝑚) =
𝑛

(𝑛,   𝑚)
  ,    𝑤ℎ𝑒𝑟𝑒 (𝑛,   𝑚)𝑑𝑒𝑛𝑜𝑡𝑒𝑠 𝑡ℎ𝑒 𝐻. 𝐶. 𝐹 𝑜𝑓 𝑛 𝑎𝑛𝑑 𝑚 

To determine 𝑜(𝑎2) 



842 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

Here, (105,   2) = 𝐻. 𝐶. 𝐹 𝑜𝑓 105 𝑎𝑛𝑑 2  ∴ 𝑜(𝑎2) =
105

(105,   2)
=

105

1
= 105 ⇒ 𝑜(𝑎2) = 105  

To determine 𝑜(𝑎3) 

Here, (105,   3) = 𝐻. 𝐶. 𝐹 𝑜𝑓 105 𝑎𝑛𝑑 3  ∴ 𝑜(𝑎3) =
105

(105,   3)
=

105

3
= 35 ⇒ 𝑜(𝑎3) = 35 

To determine 𝑜(𝑎4) 

Here, (105,   4) = 𝐻. 𝐶. 𝐹 𝑜𝑓 105 𝑎𝑛𝑑 4  ∴ 𝑜(𝑎4) =
105

(105,   4)
=

105

1
= 105 ⇒ 𝑜(𝑎4) = 105  

To determine 𝑜(𝑎5) 

Here, (105,   5) = 𝐻. 𝐶. 𝐹 𝑜𝑓 105 𝑎𝑛𝑑 5  ∴ 𝑜(𝑎5) =
105

(105,   5)
=

105

5
= 21 ⇒ 𝑜(𝑎5) = 21  

To determine 𝑜(𝑎6) 

Here, (105,   6) = 𝐻. 𝐶. 𝐹 𝑜𝑓 105 𝑎𝑛𝑑 6  ∴ 𝑜(𝑎6) =
105

(105,   6)
=

105

1
= 105 ⇒ 𝑜(𝑎6) = 105  

To determine 𝑜(𝑎7) 

Here, (107,   7) = 𝐻. 𝐶. 𝐹 𝑜𝑓 107 𝑎𝑛𝑑 7  ∴ 𝑜(𝑎7) =
105

(105,   7)
=

105

7
= 15 ⇒ 𝑜(𝑎7) = 15  

To determine 𝑜(𝑎8) 

Here, (105,   8) = 𝐻. 𝐶. 𝐹 𝑜𝑓 105 𝑎𝑛𝑑 8  ∴ 𝑜(𝑎8) =
105

(105,   8)
=

105

1
= 105 ⇒ 𝑜(𝑎8) = 105  

To determine 𝑜(𝑎9) 

Here, (105,   9) = 𝐻. 𝐶. 𝐹 𝑜𝑓 105 𝑎𝑛𝑑 9  ∴ 𝑜(𝑎9) =
105

(105,   9)
=

105

3
= 35 ⇒ 𝑜(𝑎9) = 35  

To determine 𝑜(𝑎10) 

Here, (105,   10) = 𝐻. 𝐶. 𝐹 𝑜𝑓 105 𝑎𝑛𝑑 10  ∴ 𝑜(𝑎10) =
105

(105,   10)
=

105

5
= 21 ⇒ 𝑜(𝑎10) = 21  

To determine 𝑜(𝑎11): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎11) =
105

(105,   11)
=

105

1
= 105 ⇒ 𝑜(𝑎11) = 105  

To determine 𝑜(𝑎12): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎12) =
105

(105,   12)
=

105

3
= 35 ⇒ 𝑜(𝑎12) = 35  

To determine 𝑜(𝑎13): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎13) =
105

(105,   13)
=

105

1
= 105 ⇒ 𝑜(𝑎13) = 105  

To determine 𝑜(𝑎14): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎14) =
105

(105,   14)
=

105

7
= 15 ⇒ 𝑜(𝑎14) = 15  

To determine 𝑜(𝑎15): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎15) =
105

(105,   15)
=

105

15
= 7 ⇒ 𝑜(𝑎15) = 7 

To determine 𝑜(𝑎16): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎16) =
105

(105,   16)
=

105

1
= 105 ⇒ 𝑜(𝑎16) = 105  

To determine 𝑜(𝑎17):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎17) =
105

(105,   17)
=

105

1
= 105 ⇒ 𝑜(𝑎17) = 105 

To determine 𝑜(𝑎18): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎18) =
105

(105,   18)
=

105

3
= 35 ⇒ 𝑜(𝑎18) = 35  

To determine 𝑜(𝑎19): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎19) =
105

(105,   19)
=

105

1
= 105 ⇒ 𝑜(𝑎19) = 105  

To determine 𝑜(𝑎20): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎20) =
105

(105,   20)
=

105

5
= 21 ⇒ 𝑜(𝑎20) = 21 

To determine 𝑜(𝑎21): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎21) =
105

(105,   21)
=

105

21
= 5 ⇒ 𝑜(𝑎21) = 5  

To determine 𝑜(𝑎22): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎12) =
105

(105,   22)
=

105

1
= 105 ⇒ 𝑜(𝑎22) = 105  

To determine 𝑜(𝑎23): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎23) =
105

(105,   23)
=

105

1
= 105 ⇒ 𝑜(𝑎23) = 105  

To determine 𝑜(𝑎24): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎24) =
105

(105,   24)
=

105

3
= 35 ⇒ 𝑜(𝑎24) = 35  

To determine 𝑜(𝑎25): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎25) =
105

(105,   25)
=

105

5
= 21 ⇒ 𝑜(𝑎25) = 21 

To determine 𝑜(𝑎26):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎26) =
105

(105,   26)
=

105

1
= 105 ⇒ 𝑜(𝑎26) = 105  



843 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

To determine 𝑜(𝑎27): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎27) =
105

(105,   27)
=

105

3
= 35 ⇒ 𝑜(𝑎27) = 35 

To determine 𝑜(𝑎28): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎28) =
105

(105,   28)
=

105

7
= 15 ⇒ 𝑜(𝑎28) = 15  

To determine 𝑜(𝑎29): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎29) =
105

(105,   29)
=

105

1
= 105 ⇒ 𝑜(𝑎29) = 105  

To determine 𝑜(𝑎30): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎30) =
105

(105,   30)
=

105

15
= 7 ⇒ 𝑜(𝑎30) = 7 

To determine 𝑜(𝑎31): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎31) =
105

(105,   31)
=

105

1
= 105 ⇒ 𝑜(𝑎29) = 105  

To determine 𝑜(𝑎32): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎32) =
105

(105,   32)
=

105

1
= 105 ⇒ 𝑜(𝑎32) = 105  

To determine 𝑜(𝑎33): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎33) =
105

(105,   33)
=

105

3
= 35 ⇒ 𝑜(𝑎33) = 35  

To determine 𝑜(𝑎34): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎34) =
105

(105,   34)
=

105

1
= 105 ⇒ 𝑜(𝑎34) = 105 

To determine 𝑜(𝑎35): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎35) =
105

(105,   35)
=

105

35
= 7 ⇒ 𝑜(𝑎35) = 7 

To determine 𝑜(𝑎36): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎36) =
105

(105,   36)
=

105

3
= 35 ⇒ 𝑜(𝑎36) = 35  

To determine 𝑜(𝑎37): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎37) =
105

(105,   37)
=

105

1
= 105 ⇒ 𝑜(𝑎37) = 105  

To determine 𝑜(𝑎38): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎38) =
105

(105,   38)
=

105

1
= 105 ⇒ 𝑜(𝑎38) = 105  

To determine 𝑜(𝑎39): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎39) =
105

(105,   39)
=

105

3
= 35 ⇒ 𝑜(𝑎39) = 35 

To determine 𝑜(𝑎40): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎40) =
105

(105,   40)
=

105

5
= 21 ⇒ 𝑜(𝑎40) = 21  

To determine 𝑜(𝑎41): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎41) =
105

(105,   41)
=

105

1
= 105 ⇒ 𝑜(𝑎41) = 105  

To determine 𝑜(𝑎42): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎42) =
105

(105,   42)
=

105

21
= 5 ⇒ 𝑜(𝑎41) = 5  

To determine 𝑜(𝑎43): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎43) =
105

(105,   43)
=

105

1
= 105 ⇒ 𝑜(𝑎43) = 105  

To determine 𝑜(𝑎44): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎44) =
105

(105,   44)
=

105

1
= 105 ⇒ 𝑜(𝑎44) = 105  

To determine 𝑜(𝑎45): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎45) =
105

(105,   45)
=

105

15
= 7 ⇒ 𝑜(𝑎45) = 7 

To determine 𝑜(𝑎46): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎47) =
105

(105,   46)
=

105

1
= 105 ⇒ 𝑜(𝑎46) = 105 

To determine 𝑜(𝑎47): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎47) =
105

(105,   47)
=

105

1
= 105 ⇒ 𝑜(𝑎47) = 105 

To determine 𝑜(𝑎48): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎48) =
105

(105,   48)
=

105

3
= 35 ⇒ 𝑜(𝑎48) = 35  

To determine 𝑜(𝑎49): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎49) =
105

(105,   49)
=

105

7
= 15 ⇒ 𝑜(𝑎49) = 15  

To determine 𝑜(𝑎50): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎50) =
105

(105,   50)
=

105

5
= 21 ⇒ 𝑜(𝑎50) = 21  

To determine 𝑜(𝑎51): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎51) =
105

(105,   51)
=

105

3
= 35 ⇒ 𝑜(𝑎51) = 35 

To determine 𝑜(𝑎52): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎52) =
105

(105,   52)
=

105

1
= 105 ⇒ 𝑜(𝑎52) = 105 

To determine 𝑜(𝑎53): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎53) =
105

(105,   53)
=

105

1
= 105 ⇒ 𝑜(𝑎53) = 105  

To determine 𝑜(𝑎54): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎54) =
105

(105,   54)
=

105

3
= 35 ⇒ 𝑜(𝑎54) = 35  

To determine 𝑜(𝑎55): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎55) =
105

(105,   55)
=

105

5
= 21 ⇒ 𝑜(𝑎55) = 21  

To determine 𝑜(𝑎56): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎56) =
105

(105,   56)
=

105

7
= 15 ⇒ 𝑜(𝑎56) = 15  



844 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

To determine 𝑜(𝑎57): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎57) =
105

(105,   57)
=

105

3
= 105 ⇒ 𝑜(𝑎57) = 105  

To determine 𝑜(𝑎58): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎58) =
105

(105,   58)
=

105

1
= 105 ⇒ 𝑜(𝑎58) = 105  

To determine 𝑜(𝑎59): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎59) =
105

(105,   59)
=

105

1
= 105 ⇒ 𝑜(𝑎59) = 105  

To determine 𝑜(𝑎60): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎60) =
105

(105,   60)
=

105

15
= 7 ⇒ 𝑜(𝑎60) = 7  

To determine 𝑜(𝑎61): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎61) =
105

(105,   61)
=

105

1
= 105 ⇒ 𝑜(𝑎61) = 105   

To determine 𝑜(𝑎62): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎62) =
105

(105,   62)
=

105

1
= 105 ⇒ 𝑜(𝑎62) = 105  

To determine 𝑜(𝑎63): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎63) =
105

(105,   63)
=

105

21
= 5 ⇒ 𝑜(𝑎63) = 5   

To determine 𝑜(𝑎64): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎64) =
105

(105,   64)
=

105

1
= 105 ⇒ 𝑜(𝑎64) = 105   

To determine 𝑜(𝑎65): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎65) =
105

(105,   65)
=

105

5
= 21 ⇒ 𝑜(𝑎65) = 21   

To determine 𝑜(𝑎66): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎66) =
105

(105,   66)
=

105

3
= 35 ⇒ 𝑜(𝑎66) = 35   

To determine 𝑜(𝑎67): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎67) =
105

(105,   67)
=

105

1
= 105 ⇒ 𝑜(𝑎67) = 105 

To determine 𝑜(𝑎68): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎68) =
105

(105,   68)
=

105

1
= 105 ⇒ 𝑜(𝑎68) = 105   

To determine 𝑜(𝑎69): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎69) =
105

(105,   69)
=

105

3
= 105 ⇒ 𝑜(𝑎69) = 105   

To determine 𝑜(𝑎70): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎70) =
105

(105,   70)
=

105

35
= 3 ⇒ 𝑜(𝑎70) = 3  

To determine 𝑜(𝑎71): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎71) =
105

(105,   71)
=

105

1
= 105 ⇒ 𝑜(𝑎71) = 105   

To determine 𝑜(𝑎72): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎72) =
105

(105,   72)
=

105

3
= 35 ⇒ 𝑜(𝑎72) = 35 

To determine 𝑜(𝑎73): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎73) =
105

(105,   73)
=

105

1
= 105 ⇒ 𝑜(𝑎73) = 105   

To determine 𝑜(𝑎74): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎74) =
105

(105,   74)
=

105

1
= 105 ⇒ 𝑜(𝑎74) = 105   

To determine 𝑜(𝑎75): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎75) =
105

(105,   75)
=

105

15
= 7 ⇒ 𝑜(𝑎75) = 7   

To determine 𝑜(𝑎76): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎76) =
105

(105,   76)
=

105

1
= 105 ⇒ 𝑜(𝑎76) = 105   

To determine 𝑜(𝑎77): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎77) =
105

(105,   77)
=

105

7
= 15 ⇒ 𝑜(𝑎77) = 15 

To determine 𝑜(𝑎78): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎78) =
105

(105,   78)
=

105

3
= 35 ⇒ 𝑜(𝑎78) = 35   

To determine 𝑜(𝑎79): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎79) =
105

(105,   79)
=

105

1
= 105 ⇒ 𝑜(𝑎79) = 105   

To determine 𝑜(𝑎80): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎80) =
105

(105,   80)
=

105

5
= 21 ⇒ 𝑜(𝑎80) = 21 

To determine 𝑜(𝑎81): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎81) =
105

(105,   81)
=

105

3
= 35 ⇒ 𝑜(𝑎81) = 35   

To determine 𝑜(𝑎82): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎82) =
105

(105,   82)
=

105

1
= 105 ⇒ 𝑜(𝑎82) = 105 

To determine 𝑜(𝑎83): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎83) =
105

(105,   83)
=

105

1
= 105 ⇒ 𝑜(𝑎83) = 105   

To determine 𝑜(𝑎84): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎84) =
105

(105,   84)
=

105

1
= 105 ⇒ 𝑜(𝑎84) = 105   

To determine 𝑜(𝑎85): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎85) =
105

(105,   85)
=

105

5
= 21 ⇒ 𝑜(𝑎85) = 21  

To determine 𝑜(𝑎86): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎86) =
105

(105,   86)
=

105

1
= 105 ⇒ 𝑜(𝑎86) = 105  



845 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

To determine 𝑜(𝑎87): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎87) =
105

(105,   87)
=

105

3
= 35 ⇒ 𝑜(𝑎87) = 35 

To determine 𝑜(𝑎88): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎88) =
105

(105,   88)
=

105

1
= 105 ⇒ 𝑜(𝑎88) = 105   

To determine 𝑜(𝑎89): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎89) =
105

(105,   89)
=

105

1
= 105 ⇒ 𝑜(𝑎89) = 105 

To determine 𝑜(𝑎90): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎90) =
105

(105,   90)
=

105

15
= 7 ⇒ 𝑜(𝑎90) = 7 

To determine 𝑜(𝑎91): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎91) =
105

(105,   91)
=

105

1
= 105 ⇒ 𝑜(𝑎91) = 105   

To determine 𝑜(𝑎92): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎92) =
105

(105,   92)
=

105

1
= 105 ⇒ 𝑜(𝑎92) = 105 

To determine 𝑜(𝑎93): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎93) =
105

(105,   93)
=

105

3
= 31 ⇒ 𝑜(𝑎93) = 21   

To determine 𝑜(𝑎94): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎94) =
105

(105,   94)
=

105

1
= 105 ⇒ 𝑜(𝑎94) = 105   

To determine 𝑜(𝑎95): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎95) =
105

(105,   95)
=

105

5
= 21 ⇒ 𝑜(𝑎95) = 21   

To determine 𝑜(𝑎96): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎96) =
105

(105,   96)
=

105

3
= 35 ⇒ 𝑜(𝑎96) = 35   

To determine 𝑜(𝑎97): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎97) =
105

(105,   97)
=

105

1
= 105 ⇒ 𝑜(𝑎97) = 105 

To determine 𝑜(𝑎98): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎98) =
105

(105,   98)
=

105

7
= 15 ⇒ 𝑜(𝑎98) = 15   

To determine 𝑜(𝑎99): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎99) =
105

(105,   99)
=

105

3
= 35 ⇒ 𝑜(𝑎99) = 35 

To determine 𝑜(𝑎100): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎100) =
105

(105,   100)
=

105

5
= 21 ⇒ 𝑜(𝑎100) = 21 

To determine 𝑜(𝑎101): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎100) =
105

(105,   100)
=

105

5
= 21 ⇒ 𝑜(𝑎100) = 21 

To determine 𝑜(𝑎102): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎102) =
105

(105,   102)
=

105

3
= 35 ⇒ 𝑜(𝑎102) = 35 

To determine 𝑜(𝑎103): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎103) =
105

(105,   103)
=

105

21
= 5 ⇒ 𝑜(𝑎103) = 5 

To determine 𝑜(𝑎104): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎104) =
105

(105,   104)
=

105

1
= 105 ⇒ 𝑜(𝑎104) = 105 

To determine 𝑜(𝑎105): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎105) =
105

(105,   105)
=

105

105
= 1 ⇒ 𝑜(𝑎100) = 1 

 
4.3. The Higher 107 Order of a Group for Multiplication Composition [14] 

Find the order of every element in the multiplication group 𝐺 = {𝑎, 𝑎2, 𝑎3, 𝑎4,   . . . , 𝑎107 = 𝑒} 
Solution: 

The identity element of the given group is 𝑎107 = 𝑒 ⇒ 𝑜(𝑎) = 107  ∴  𝑜(𝑎) = 107   

We know that 𝑜(𝑎𝑚) =
𝑛

(𝑛,   𝑚)
  ,    𝑤ℎ𝑒𝑟𝑒 (𝑛,   𝑚)𝑑𝑒𝑛𝑜𝑡𝑒𝑠 𝑡ℎ𝑒 𝐻. 𝐶. 𝐹 𝑜𝑓 𝑛 𝑎𝑛𝑑 𝑚 

To determine 𝑜(𝑎2) 

Here, (107,   2) = 𝐻. 𝐶. 𝐹 𝑜𝑓 107 𝑎𝑛𝑑 2  ∴ 𝑜(𝑎2) =
107

(107,   2)
=

107

1
= 107 ⇒ 𝑜(𝑎2) = 107  

To determine 𝑜(𝑎3) 

Here, (107,   3) = 𝐻. 𝐶. 𝐹 𝑜𝑓 107 𝑎𝑛𝑑 3  ∴ 𝑜(𝑎3) =
107

(107,   3)
=

107

1
= 107 ⇒ 𝑜(𝑎3) = 107 

To determine 𝑜(𝑎4) 

Here, (107,   4) = 𝐻. 𝐶. 𝐹 𝑜𝑓 107 𝑎𝑛𝑑 4  ∴ 𝑜(𝑎4) =
107

(107,   4)
=

107

1
= 107 ⇒ 𝑜(𝑎4) = 107 

To determine 𝑜(𝑎5) 

Here, (107,   5) = 𝐻. 𝐶. 𝐹 𝑜𝑓 107 𝑎𝑛𝑑 5  ∴ 𝑜(𝑎5) =
107

(107,   5)
=

107

1
= 107 ⇒ 𝑜(𝑎5) = 107 



846 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

To determine 𝑜(𝑎6) 

Here, (107,   6) = 𝐻. 𝐶. 𝐹 𝑜𝑓 107 𝑎𝑛𝑑 6  ∴ 𝑜(𝑎3) =
107

(107,   6)
=

107

1
= 107 ⇒ 𝑜(𝑎6) = 107 

To determine 𝑜(𝑎7) 

Here, (107,   7) = 𝐻. 𝐶. 𝐹 𝑜𝑓 107 𝑎𝑛𝑑 7  ∴ 𝑜(𝑎7) =
107

(107,   7)
=

107

1
= 107 ⇒ 𝑜(𝑎7) = 107  

To determine 𝑜(𝑎8) 

Here, (107,   8) = 𝐻. 𝐶. 𝐹 𝑜𝑓 107 𝑎𝑛𝑑 8  ∴ 𝑜(𝑎8) =
107

(107,   8)
=

107

1
= 107 ⇒ 𝑜(𝑎8) = 107 

To determine 𝑜(𝑎9) 

Here, (107,   9) = 𝐻. 𝐶. 𝐹 𝑜𝑓 107 𝑎𝑛𝑑 9  ∴ 𝑜(𝑎9) =
107

(107,   9)
=

107

1
= 107 ⇒ 𝑜(𝑎9) = 107 

To determine 𝑜(𝑎10): 𝑆𝑜 𝑡ℎ𝑎𝑡𝑜(𝑎10) =
107

(107,   10)
=

107

1
= 107 ⇒ 𝑜(𝑎10) = 107 

To determine 𝑜(𝑎11): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎11) =
107

(107,   11)
=

107

1
= 107 ⇒ 𝑜(𝑎11) = 107 

To determine 𝑜(𝑎12): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎13) =
107

(107,   12)
=

107

1
= 107 ⇒ 𝑜(𝑎12) = 107 

To determine 𝑜(𝑎13): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎13) =
107

(107,   13)
=

107

1
= 107 ⇒ 𝑜(𝑎13) = 107  

To determine 𝑜(𝑎14): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎14) =
107

(107,   14)
=

107

1
= 107 ⇒ 𝑜(𝑎14) = 107  

To determine 𝑜(𝑎15): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎15) =
107

(107,   15)
=

107

1
= 107 ⇒ 𝑜(𝑎15) = 107 

To determine 𝑜(𝑎16): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎16) =
107

(107,   16)
=

107

1
= 107 ⇒ 𝑜(𝑎16) = 107  

To determine 𝑜(𝑎17): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎17) =
107

(107,   17)
=

107

1
= 107 ⇒ 𝑜(𝑎17) = 107 

To determine 𝑜(𝑎18): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎18) =
107

(107,   18)
=

107

1
= 107 ⇒ 𝑜(𝑎18) = 107  

To determine 𝑜(𝑎19): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎19) =
107

(107,   19)
=

107

1
= 107 ⇒ 𝑜(𝑎19) = 107  

To determine 𝑜(𝑎20): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎20) =
107

(107,   20)
=

107

1
= 107 ⇒ 𝑜(𝑎20) = 107 

To determine 𝑜(𝑎21): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎21) =
107

(107,   21)
=

107

1
= 107 ⇒ 𝑜(𝑎21) = 107  

To determine 𝑜(𝑎22): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎22) =
107

(107,   22)
=

107

1
= 107 ⇒ 𝑜(𝑎22) = 107  

To determine 𝑜(𝑎23): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎23) =
107

(107,   23)
=

107

1
= 107 ⇒ 𝑜(𝑎23) = 107  

To determine 𝑜(𝑎24): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎24) =
107

(107,   24)
=

107

1
= 107 ⇒ 𝑜(𝑎24) = 107  

To determine 𝑜(𝑎25): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎25) =
107

(107,   25)
=

107

1
= 107 ⇒ 𝑜(𝑎25) = 107 

To determine 𝑜(𝑎26): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎26) =
107

(107,   26)
=

107

1
= 107 ⇒ 𝑜(𝑎26) = 107  

To determine 𝑜(𝑎27): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎27) =
107

(107,   27)
=

107

1
= 107 ⇒ 𝑜(𝑎27) = 107 

To determine 𝑜(𝑎28): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎28) =
107

(107,   28)
=

107

1
= 107 ⇒ 𝑜(𝑎28) = 107  

To determine 𝑜(𝑎29): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎29) =
107

(107,   29)
=

107

1
= 107 ⇒ 𝑜(𝑎29) = 107  

To determine 𝑜(𝑎30): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎30) =
107

(107,   30)
=

107

1
= 107 ⇒ 𝑜(𝑎30) = 107 

To determine 𝑜(𝑎31): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎31) =
107

(107,   31)
=

107

1
= 107 ⇒ 𝑜(𝑎31) = 107 

To determine 𝑜(𝑎32): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎32) =
107

(107,   32)
=

107

1
= 107 ⇒ 𝑜(𝑎32) = 107 



847 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

To determine 𝑜(𝑎33): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎33) =
107

(107,   33)
=

107

1
= 107 ⇒ 𝑜(𝑎33) = 107  

To determine 𝑜(𝑎34): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎34) =
107

(107,   34)
=

107

1
= 107 ⇒ 𝑜(𝑎34) = 107 

To determine 𝑜(𝑎35): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎31) =
107

(107,   35)
=

107

1
= 107 ⇒ 𝑜(𝑎35) = 107 

To determine 𝑜(𝑎36): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎36) =
107

(107,   36)
=

107

1
= 107 ⇒ 𝑜(𝑎36) = 107  

To determine 𝑜(𝑎37): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎37) =
107

(107,   37)
=

107

1
= 107 ⇒ 𝑜(𝑎37) = 107  

To determine 𝑜(𝑎38): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎38) =
107

(107,   38)
=

107

1
= 107 ⇒ 𝑜(𝑎38) = 107  

To determine 𝑜(𝑎39): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎39) =
107

(107,   39)
=

107

1
= 107 ⇒ 𝑜(𝑎39) = 107 

To determine 𝑜(𝑎40): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎40) =
107

(107,   40)
=

107

1
= 107 ⇒ 𝑜(𝑎40) = 107  

To determine 𝑜(𝑎41): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎41) =
107

(107,   41)
=

107

1
= 107 ⇒ 𝑜(𝑎41) = 107   

To determine 𝑜(𝑎42): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎42) =
107

(107,   42)
=

107

1
= 107 ⇒ 𝑜(𝑎42) = 107 

To determine 𝑜(𝑎43): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎43) =
107

(107,   43)
=

107

1
= 107 ⇒ 𝑜(𝑎43) = 107 

To determine 𝑜(𝑎44): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎44) =
107

(107,   44)
=

107

1
= 107 ⇒ 𝑜(𝑎44) = 107   

To determine 𝑜(𝑎45): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎45) =
107

(107,   45)
=

107

1
= 107 ⇒ 𝑜(𝑎45) = 107 

To determine 𝑜(𝑎46): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎46) =
107

(107,   46)
=

107

1
= 107 ⇒ 𝑜(𝑎46) = 107 

To determine 𝑜(𝑎47): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎47) =
107

(107,   47)
=

107

1
= 107 ⇒ 𝑜(𝑎47) = 107 

To determine 𝑜(𝑎48): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎48) =
107

(107,   48)
=

107

1
= 107 ⇒ 𝑜(𝑎48) = 107   

To determine 𝑜(𝑎49): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎49) =
107

(107,   49)
=

107

1
= 107 ⇒ 𝑜(𝑎49) = 107   

To determine 𝑜(𝑎50): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎50) =
107

(107,   50)
=

107

1
= 107 ⇒ 𝑜(𝑎50) = 107   

To determine 𝑜(𝑎51): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎51) =
107

(107,   51)
=

107

1
= 107 ⇒ 𝑜(𝑎51) = 107 

To determine 𝑜(𝑎52): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎52) =
107

(107,   52)
=

107

1
= 107 ⇒ 𝑜(𝑎52) = 107 

To determine 𝑜(𝑎53): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎53) =
107

(107,   53)
=

107

1
= 107 ⇒ 𝑜(𝑎53) = 107  

To determine 𝑜(𝑎54): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎54) =
107

(107,   54)
=

107

1
= 107 ⇒ 𝑜(𝑎54) = 107  

To determine 𝑜(𝑎55): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎55) =
107

(107,   55)
=

107

1
= 107 ⇒ 𝑜(𝑎55) = 107  

To determine 𝑜(𝑎56): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎56) =
107

(107,   56)
=

107

1
= 107 ⇒ 𝑜(𝑎56) = 107  

To determine 𝑜(𝑎57): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎57) =
107

(107,   57)
=

107

1
= 107 ⇒ 𝑜(𝑎57) = 107  

To determine 𝑜(𝑎58): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎50) =
107

(107,   58)
=

107

1
= 107 ⇒ 𝑜(𝑎58) = 107  

To determine 𝑜(𝑎59): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎59) =
107

(107,   59)
=

107

1
= 107 ⇒ 𝑜(𝑎59) = 107  

To determine 𝑜(𝑎60): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎60) =
107

(107,   60)
=

107

1
= 107 ⇒ 𝑜(𝑎60) = 107 

To determine 𝑜(𝑎61): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎61) =
107

(107,   61)
=

107

1
= 107 ⇒ 𝑜(𝑎61) = 107   

To determine 𝑜(𝑎62): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎62) =
107

(107,   62)
=

107

1
= 107 ⇒ 𝑜(𝑎62) = 107  



848 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

To determine 𝑜(𝑎63): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎63) =
107

(107,   63)
=

107

1
= 107 ⇒ 𝑜(𝑎63) = 107   

To determine 𝑜(𝑎64): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎64) =
107

(107,   64)
=

107

1
= 107 ⇒ 𝑜(𝑎64) = 107   

To determine 𝑜(𝑎65): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎65) =
107

(107,   65)
=

107

1
= 107 ⇒ 𝑜(𝑎65) = 107   

To determine 𝑜(𝑎66):𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎66) =
107

(107,   66)
=

107

1
= 107 ⇒ 𝑜(𝑎66) = 107   

To determine 𝑜(𝑎67): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎67) =
105

(105,   67)
=

105

1
= 105 ⇒ 𝑜(𝑎67) = 105 

To determine 𝑜(𝑎68): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎68) =
107

(107,   68)
=

107

1
= 107 ⇒ 𝑜(𝑎68) = 107   

To determine 𝑜(𝑎69): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎69) =
107

(107,   69)
=

107

1
= 107 ⇒ 𝑜(𝑎69) = 107   

To determine 𝑜(𝑎70): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎70) =
107

(107,   70)
=

107

1
= 107 ⇒ 𝑜(𝑎70) = 107 

To determine 𝑜(𝑎71): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎71) =
107

(107,   71)
=

107

1
= 107 ⇒ 𝑜(𝑎71) = 107   

To determine 𝑜(𝑎72): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎72) =
107

(107,   72)
=

107

1
= 107 ⇒ 𝑜(𝑎72) = 107 

To determine 𝑜(𝑎73): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎73) =
107

(107,   73)
=

107

1
= 107 ⇒ 𝑜(𝑎73) = 107   

To determine 𝑜(𝑎74): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎74) =
107

(107,   74)
=

107

1
= 107 ⇒ 𝑜(𝑎74) = 107   

To determine 𝑜(𝑎75): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎75) =
107

(107,   75)
=

107

1
= 107 ⇒ 𝑜(𝑎75) = 107   

To determine 𝑜(𝑎76): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎76) =
107

(107,   76)
=

107

1
= 107 ⇒ 𝑜(𝑎76) = 107   

To determine 𝑜(𝑎77): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎77) =
107

(107,   77)
=

107

1
= 107 ⇒ 𝑜(𝑎77) = 107 

To determine 𝑜(𝑎78): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎78) =
107

(107,   78)
=

107

1
= 107 ⇒ 𝑜(𝑎78) = 107   

To determine 𝑜(𝑎79): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎79) =
107

(107,   79)
=

107

1
= 107 ⇒ 𝑜(𝑎79) = 107   

To determine 𝑜(𝑎80): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎80) =
107

(107,   80)
=

107

1
= 107 ⇒ 𝑜(𝑎80) = 107 

To determine 𝑜(𝑎81): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎81) =
107

(107,   81)
=

107

1
= 107 ⇒ 𝑜(𝑎81) = 107   

To determine 𝑜(𝑎82): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎82) =
107

(107,   82)
=

107

1
= 107 ⇒ 𝑜(𝑎82) = 107 

To determine 𝑜(𝑎83): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎83) =
107

(107,   83)
=

107

1
= 107 ⇒ 𝑜(𝑎83) = 107   

To determine 𝑜(𝑎84): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎84) =
107

(107,   84)
=

107

1
= 107 ⇒ 𝑜(𝑎84) = 107   

To determine 𝑜(𝑎85): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎85) =
107

(107,   85)
=

107

1
= 107 ⇒ 𝑜(𝑎85) = 107  

To determine 𝑜(𝑎86): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎86) =
107

(107,   86)
=

107

1
= 107 ⇒ 𝑜(𝑎86) = 107  

To determine 𝑜(𝑎87): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎87) =
107

(107,   87)
=

107

1
= 107 ⇒ 𝑜(𝑎87) = 107 

To determine 𝑜(𝑎88): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎88) =
107

(107,   88)
=

107

1
= 107 ⇒ 𝑜(𝑎88) = 107   

To determine 𝑜(𝑎89): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎89) =
107

(107,   89)
=

107

1
= 107 ⇒ 𝑜(𝑎89) = 107 

To determine 𝑜(𝑎90): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎90) =
107

(107,   90)
=

107

1
= 107 ⇒ 𝑜(𝑎90) = 107 

To determine 𝑜(𝑎91): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎91) =
107

(107,   91)
=

107

1
= 107 ⇒ 𝑜(𝑎91) = 107   

To determine 𝑜(𝑎92): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎92) =
107

(107,   92)
=

107

1
= 107 ⇒ 𝑜(𝑎92) = 107 



849 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

To determine 𝑜(𝑎93): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎93) =
107

(107,   93)
=

107

1
= 107 ⇒ 𝑜(𝑎93) = 107   

To determine 𝑜(𝑎94): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎94) =
107

(107,   94)
=

107

1
= 107 ⇒ 𝑜(𝑎94) = 107   

To determine 𝑜(𝑎95): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎95) =
107

(107,   95)
=

107

1
= 107 ⇒ 𝑜(𝑎95) = 107   

To determine 𝑜(𝑎96): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎96) =
107

(107,   96)
=

107

1
= 107 ⇒ 𝑜(𝑎96) = 107   

To determine 𝑜(𝑎97): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎97) =
107

(107,   97)
=

107

1
= 107 ⇒ 𝑜(𝑎97) = 107 

To determine 𝑜(𝑎98): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎98) =
107

(107,   98)
=

107

1
= 107 ⇒ 𝑜(𝑎98) = 107   

To determine 𝑜(𝑎99): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎99) =
107

(107,   99)
=

107

1
= 107 ⇒ 𝑜(𝑎99) = 107 

To determine 𝑜(𝑎100): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎100) =
107

(107,   100)
=

107

1
= 107 ⇒ 𝑜(𝑎100) = 107 

To determine 𝑜(𝑎101): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎101) =
107

(107,   101)
=

107

1
= 107 ⇒ 𝑜(𝑎101) = 107 

To determine 𝑜(𝑎102): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎102) =
107

(107,   102)
=

107

1
= 107 ⇒ 𝑜(𝑎102) = 107 

To determine 𝑜(𝑎103): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎103) =
107

(107,   103)
=

107

1
= 107 ⇒ 𝑜(𝑎103) = 107 

To determine 𝑜(𝑎104): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎104) =
107

(107,   104)
=

107

1
= 107 ⇒ 𝑜(𝑎104) = 107 

To determine 𝑜(𝑎105): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎105) =
107

(107,   105)
=

107

1
= 107 ⇒ 𝑜(𝑎105) = 107 

To determine 𝑜(𝑎106): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎106) =
107

(107,   106)
=

107

1
= 107 ⇒ 𝑜(𝑎106) = 107 

To determine 𝑜(𝑎107): 𝑆𝑜 𝑡ℎ𝑎𝑡 𝑜(𝑎107) =
107

(107,   107)
=

107

107
= 1 ⇒ 𝑜(𝑎107) = 1 

 
5. Analysis of the Result 

We discussed the result of order of every element for multiplication composition in the higher 100, 
105 and 107 orders of group for multiplication composition. In fact, we can use composition related 

theorem to evaluate order of group of different orders such as order 2, 3, 4, 5, . . . ,20etc., i.e., whose order 
is not so high (Not Higher Order Groups). As a result, we use multiplication related theorems to 
evaluate the order of groups of the higher order of group for composition. Here, to find orders of 
elements of a cyclic group. Then in order to find out the order of an element am in the group G.  If we 

apply this method 𝑜(𝑎𝑚) =
𝑛

(𝑛,   𝑚)
  ,  where (n, m) denotes the H.C.F of n and m, then we can easily find 

out any higher order of the group. Thus, it has been found necessary and convenient to work or solve 
these structures in details. 
 
6. Conclusion 

In this work we developed the higher order of elements of a group for various higher orders of a 
group. This result is very important for the order of every element of a group, where using 

the 𝐻. 𝐶. 𝐹 𝑜𝑓 𝑚 𝑎𝑛𝑑 𝑛 . This reason we can find out the order of elements of a group of different orders 
of a group. In different situations, once it was found that a given solution satisfies the basic result of one 
structure, and having known the properties of that structure, it becomes extremely easy to forecast the 
behavior of the situation. This result in this paper will be advantages for group theory related to 
subgroups and order of elements of a group. Thus, the demands of the work of mathematical problems 
as like as physical problems such as groups, number systems, vectors, matrices and so on. 
 
 
 



850 

 

 

Edelweiss Applied Science and Technology 
ISSN: 2576-8484   

Vol. 9, No. 7: 835-850, 2025 
DOI: 10.55214/25768484.v9i7.8743 
© 2025 by the authors; licensee Learning Gate 

 

Transparency:  
The authors confirm that the manuscript is an honest, accurate,  and  transparent  account  of  the  
study; that  no  vital  features  of  the  study  have  been  omitted;  and  that  any  discrepancies  from  
the  study  as planned have been explained. This study followed all ethical practices during writing. 
 

Acknowledgements: 
I would like to thank my respectable teacher Prof. Dr. Moqbul Hossain for encouragement and valuable 
suggestions. 
 

Copyright: 
© 2025 by the authors. This open-access article is distributed under the terms and conditions of the 
Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). 
 

References 
[1] M. Hall Jr and D. Wales, "The simple group of order 604,800," Journal of Algebra, vol. 9, no. 4, pp. 417-450, 1968.  

https://doi.org/10.1016/0021-8693(68)90014-8  
[2] R. Brauer and H. F. Tuan, "On simple groups of finite order," I Bulletin of the American Mathematical Society, vol. 51, 

no. 10, pp. 756-766, 1945.  
[3] M. A. Mannan, H. Akter, and S. Mondal, "Evaluate all possible subgroups of a group of order 30 and 42 by using 

Sylow’s theorem," International Journal of Scientific & Engineering Research, vol. 12, no. 11, pp. 139–153, 2021.  
[4] M. A. Mannan, M. A. Ullah, U. K. Dey, and M. Alauddin, "A study on sylow theorems for finding out possible 

subgroups of a group in different types of order," Mathematics and Statistics, vol. 10, pp. 851-860, 2022.  
[5] M. A. Mannan, H. Akter, and M. A. Ullah, "Evaluate all the order of every element in the higher even, odd, and prime 

order of group for composition," Science and Technology Indonesia, vol. 7, no. 3, pp. 333-343, 2022.  
https://doi.org/10.26554/sti.2022.7.3.333-343 

[6] L. Finkelstein and A. Rudvalis, "The maximal subgroups of Janko's simple group of order 50, 232, 960," Journal of 
Algebra, vol. 30, no. 1-3, pp. 122-143, 1974.  

[7] J. McKay and D. Wales, "The multipliers of the simple groups of order 604,800 and 50,232,960," Journal of Algebra, 
vol. 17, pp. 262–273, 1971.  

[8] U. Dardano and S. Rinauro, "Groups with many subgroups which are commensurable with some normal subgroup," 
Advances in Group Theory and Applications, vol. 7, pp. 3-13, 2019.  

[9] S. Trefethen, "Non-Abelian composition factors of finite groups with the CUT-property," Journal of Algebra, vol. 522, 
pp. 236-242, 2019.  https://doi.org/10.1016/j.jalgebra.2018.12.002 

[10] M. A. Mannan, N. Nahar, H. Akter, M. Begum, M. A. Ullah, and S. Mustari, "Evaluate all the order of every element 
in the higher order of group for addition and multiplication composition," International Journal of Modern Nonlinear 
Theory and Application, vol. 11, pp. 11-30, 2022.  https://doi.org/10.4236/ijmnta.2022.112002 

[11] D. J. S. Robinson, A course in the theory of groups. New York: Springer-Verlag, 1982. 
[12] B. McCann, "On products of cyclic and non-abelian finite p-groups," Advances in Group Theory and Applications, vol. 9, 

pp. 5-37, 2020.  
[13] D. Gorenstein, R. Lyons, and R. Solomon, The classification of the finite simple groups, Number 8. Providence, RI: 

American Mathematical Society, 2018. 
[14] J. M. Hall, "On the number of Sylow subgroups in a finite group," Journal of Algebra, vol. 7, no. 3, pp. 363-371, 1967.  

 

 

https://creativecommons.org/licenses/by/4.0/
https://doi.org/10.1016/0021-8693(68)90014-8
https://doi.org/10.26554/sti.2022.7.3.333-343
https://doi.org/10.1016/j.jalgebra.2018.12.002
https://doi.org/10.4236/ijmnta.2022.112002