Difference between revisions of "2003 AMC 10A Problems/Problem 16"

Revision as of 22:42, 16 July 2014

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
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
All AMC 10 Problems and Solutions

@@ Line 11: / Line 11: @@
 <math>3^{2003}=(3^{4})^{500}\cdot3^{3}\equiv1^{500}\cdot27\equiv7\pmod{10}</math>
-Therefore, the units digit is <math>7 \Rightarrow C</math>
+Therefore, the units digit is <math>7 \Rightarrow\boxed{\mathrm{(C)}\ 7}</math>
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