2023 AMC 10B Problem 8

Question

What is the units digit of

2022^{2023} + 2023^{2022}?

7

1

9

5

3

Units digits of powers of

2

cycle

2, 4, 8, 6;

powers of

3

cycle

3, 9, 7, 1

Reduce each exponent modulo

4,

then add the two units digits

Only the units digits matter. Powers of

2

cycle

2, 4, 8, 6

with period

4,

and

2023 \equiv 3 \pmod 4,

so

2022^{2023}

ends in

8.

Powers of

3

cycle

3, 9, 7, 1,

and

2022 \equiv 2 \pmod 4,

so

2023^{2022}

ends in

9.

Add them:

8 + 9 = 17,

so the units digit is

7.

Therefore, the answer is A.

Problem 8 in other years