2003 AMC 10A Problem 16 — Solution | LIVE by Po-Shen Loh

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2003 AMC 10A Problem 16

Attempt Problem 16 of the 2003 AMC 10A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 10A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

16.

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

$1$

$3$

$7$

$8$

$9$

Answer: C

Concepts:units digit modular arithmetic pattern recognition

Difficulty rating: 1350

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.

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