Remainder

4^1 divided by 9, leaves a remainder of 4.
4^2 divided by 9, leaves a remainder of 7. 
4^3 divided by 9, leaves a remainder of 1. 

4^4 divided by 9, leaves a remainder of 4.
4^5 divided by 9, leaves a remainder of 7.
4^6 divided by 9, leaves a remainder of 1. 

4^(3k+1) leaves a remainder of 4
4^(3k+2) leaves a remainder of 7

4^3k leaves a remainder of 1  

What will be the remainder when 4^123 is divided by 9? 

In 3 seconds.

123=3*41

so remainder will be 1.


Trick 1-While trying to find the cycle or pattern of remainders when p^n is divided by q, just multiply the previous remainder with p to get the next value.

Work on this post the answers-
e.g  1)3^24 divided by 7
2) 5^711 divided by 11
3) 7^321 divided by 5

Concept-Remainder of a sum when it is being divided is going to be the same as the sum of the individual remainders 

e.g. 
remainder of (12+34+54+18+48+67)  when divided by 11 is same as 233 when divided by 11 which is 2.
we can also find it by another way 
12/11 rem- 1
34/11 rem- 1
54/11 rem- 10
18/11 rem- 7
48/11 rem- 4
67/11 rem- 1
total remainder  will be 1+1+10+7+4+1=24

24/11 remainder is 2.




Concept- Remainder of a product when it is being divided is going to be the same as the product of the individual remainders

e.g. 
remainder of (12*34*54)  when divided by 11 is same as 22032 when divided by 11 which is 10.
we can also find it by another way 
12/11 rem- 1
34/11 rem- 1
54/11 rem- 10
total remainder  will be 1*1*10=10


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