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2007 Cyprus MO/Lyceum/Problem 17

Problem

The last digit of the number $a=2^{2007}+3^{2007}+5^{2007}+7^{2007}$ is

$\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 6\qquad \mathrm{(E) \ } 8$

Solution

$2^{2007\bmod4}\equiv2^3\equiv8\mod10$

$3^{2007\bmod4}\equiv3^3\equiv7\mod10$

$5^{2007\bmod4}\equiv5^3\equiv5\mod10$

$7^{2007\bmod4}\equiv7^3\equiv3\mod10$

$8+7+5+3\equiv3\mod10$

See also

2007 Cyprus MO, Lyceum (Problems)
Preceded by Problem 16	Followed by Problem 18
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30

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