HMMT November, 2024

Number theory Difficulty 4.4 AIME Find the answer United States

Problem:

The number $17^{6}$ when written out in base 10 contains 8 distinct digits from $1,2, \ldots, 9$ , with no repeated digits or zeroes. Compute the missing nonzero digit.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

Observe that

$17^{6} \equiv (-1)^{6} = 1 \bmod 9$

If $x$ is the missing digit, then the digits of $17^{6}$ sum to $(1+2+\cdots+9)-x=45-x$ , so $45-x \equiv 1 \bmod 9$ . We conclude $x=8$ .

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