2003 AMC 10A Problem 16

Problem 16 of 25IntermediateNumber Theory

What is the units digit of

13^{2003}?

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Solution

The units digit of

13^{2003}

matches that of

3^{2003}.

Powers of

3

have units digits cycling

3,

9,

7,

1

with period

4.

Since

2003 = 4 \cdot 500 + 3,

the units digit is the third in the cycle, which is

7.

Thus, the correct answer is C.

Concepts: units digit · modular arithmetic · pattern recognition

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