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2003 AMC 10A Problems/Problem 16

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Problem

What is the units digit of 13^{2003}?

\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 7\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 9

Solution

13^{2003}\equiv 3^{2003}\pmod{10}

Since 3^4=81\equiv1\pmod{10}:

3^{2003}=(3^{4})^{500}\cdot3^{3}\equiv1^{500}\cdot27\equiv7\pmod{10}

Therefore, the units digit is 7 \Rightarrow C

See Also

2003 AMC 10A (Problems • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
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
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