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Berkeley Math Circle: Monthly Contest 2

Number theory Difficulty 4.3 AIME Find the answer United States

Problem:

Carl computes the number
$N=5^{555}+6^{666}+7^{777}$
and writes it in decimal notation. What is the last digit of $N$ that Carl writes?

Your answer

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

Solution

Solution:

We look at the last digit of each term.
- The last digit of $5^{\bullet}$ is always $5$ .
- The last digit of $6^{\bullet}$ is always $6$ .
- The last digit of $7^{\bullet}$ cycles $7, 9, 3, 1, 7, 9, 3, \ldots$ .

So the last digits are $5, 6, 7$ in that order. Since $5+6+7=18$ , the answer is $8$ .

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