A "Think about it" number theory question.

Number Theory Level pending

Find the smallest value n such that $164^x - n^x$ is a multiple of 18 for all positive values of x. If there are no values that satisfies it, then write 165.

The answer is 2.

1 solution

Elijah L
Mar 11, 2021

Note that for all positive integers $a$ , $b$ , and $x$ , $a-b \mid a^x - b^x$ . Because $164 \equiv 2 \pmod {18}$ , the smallest positive integer is $\boxed{2}$ .

A "Think about it" number theory question.

The answer is 2.

1 solution

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