2IIM - CAT Discussion Forum

What is the remainder when 5^23 is divided by 17?

Tried remainder theorem method, but it becomes too complicated. Help solicited. Thanks

one approach to solve this question is as follows

remainder when 5 divided by 17 is 5
remainder when 5^2 divided by 17 is 8
remainder when 5^4 divided by 17 is 8*8 = 64 = 13
remainder when 5^8 divided by 17 is 13*13 = 169 = 16
remainder when 5^16 divided by 17 is 5 = 16*16 = 256 = 1.

5^23 can be written as (5^16) * (5^4) * (5^2) * 5

Therefore, when 5^23 is divided by 17 will be the product of the remainders of 5^16, 5^4, 5^2, and 5 divided by 17.

The remainders are 1 * 13 * 8 * 5 = 520.

As 520 is greater than 17, the remainder when 520 is divided by 17 will be the remainder.

i.e., 10 is the remainder.

thanks for the explanation

is there any other shorter method?