Question

what is the remainder when 127 ^97 + 97^97 divided by 32​

Answer 1

Answer: let us put it in the language of Modular airthmatic.

127 ^ 97 + 97^97 =REM (mod 32)

127 ^ 97 (mod 32) +97^ 97 (mod 32)= REM --- ( I)

Let us solve it by applying Euler 's Theorem.

PHI(32)= 2^5 =32* (1-1\2) =16 (or phi (32)= 2^5 -2^4 =16.

Since 2 is prime. 97=1 ( mod 16).

According to Euler's theorem.

127 ^1 (mod 32)=127(mod 32) =127 (mod 32) =127 (mod 32)

97^97 = 97 (mod 32)

Rewrite (I) as : --

127 (mod32) =0=( mod32) therefore the Remainder is 0.