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number theory intermediate

Problem

What is the units digit of

1 3^{2003}

?

(A)

1

(B)

3

(C)

7

(D)

8

Solution

1 3^{2003} \equiv 3^{2003} (mod 10)

Since

3^{4} = 81 \equiv 1 (mod 10)

:

3^{2003} = (3^{4})^{500} \cdot 3^{3} \equiv 1^{500} \cdot 27 \equiv 7 (mod 10)

Therefore, the units digit is

7 \Rightarrow 7

Final answer

C