2IIM

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.

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thanks for the explanation

is there any other shorter method?