Competition problem — Number theory

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Number theory Difficulty 5.3 AIME, harder Find the answer

8. Develop a test for divisibility by 37 , based on the fact that $10^{3} \equiv 1(\bmod 37)$ . Use this to check 443692 and 11092785 for divisibility by 37.

Your answer

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

Solution

8. $a_{2 n} a_{2 n-1} \ldots a_{1} a_{0} \equiv a_{2 n} a_{2 n-1} a_{2 n-2}+\cdots+a_{5} a_{4} a_{3}+a_{2} a_{1} a_{0}(\bmod 37)$ ,
$37 \backslash 443692,37 \mid 11092785$

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.