2003 AMC 10A Problems/Problem 16

Problem

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

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

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

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 $7 \Rightarrow\boxed{\mathrm{(C)}\ 7}$

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