Competition problem — Number theory

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Number theory Difficulty 2.8 Junior Find the answer

When $5^{35}-6^{21}$ is evaluated, what is the units (ones) digit?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

First, we note that $5^{35}-6^{21}$ is a positive integer. Second, we note that any positive integer power of 5 has a units digit of 5. Similarly, each power of 6 has a units digit of 6. Therefore, $5^{35}$ has a units digit of 5 and $6^{21}$ has a units digit of 6. When a positive integer with units digit 6 is subtracted from a larger positive integer whose units digit is 5, the difference has a units digit of 9. Therefore, $5^{35}-6^{21}$ has a units digit of 9.

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