2024 AMC 12B Problem 14

Attempt Problem 14 of the 2024 AMC 12B 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 2024 AMC 12B solutions, or check the answer key.

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

14.

How many different remainders can result when the

100

th power of an integer is divided by

125?

1

2

5

25

125

Split into integers coprime to

5

and integers divisible by

5

Since

\varphi(125) = 100,

Euler’s theorem gives

n^{100} \equiv 1 \pmod{125}

when

\gcd(n,5)=1;

a multiple of

5

has

n^{100}

divisible by

125

n

is coprime to

5,

then since

\varphi(125) = 100,

Euler’s theorem gives

n^{100} \equiv 1 \pmod{125}.

n

is a multiple of

5,

then

n^{100}

is divisible by

5^{100},

hence by

125,

leaving remainder

0.

So the only possible remainders are

0

and

1,

which is

2

distinct values.

Thus, the correct answer is B.

Problem 14 in other years