The ten's digit of $9^{20}$ is 0, what is the tens' digit of $9^{19}$ ?

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Official Explanation

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If you are thorough with number properties, this will turn out to be a very easy one. Otherwise, you may have difficulty to start with.

First, consider the units digit pattern for successive powers of 9: it is 9, 1, 9, 1... For odd powers, the units digit is 9; for even powers, the units digit is 1.

9^{20}

will have units digit 1, so its last two digits are ...01.

9^{19}

will have units digit 9, so its last two digits will be ...x9.

If we multiply ...x9 by 9, we get tens digit as

9x + 8

(where 8 is the carry over from

9 \times 9 = 81

).…

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What this question tests

What it tests

Your understanding of divisibility rules, least common multiples, and greatest common factors.

Common trap

Forgetting that 1 is not prime and that every integer divides 0.

The ten's digit of 9209^{20}920 is 0, what is the tens' digit of 9199^{19}919?