1965 AMC 12 Problem 33

Attempt Problem 33 of the 1965 AMC 12 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 1965 AMC 12 solutions, or check the answer key.

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33.

If the number

15!,

that is,

15\cdot14\cdot13\cdots1,

ends with

k

zeros when given to the base

12

and ends with

h

zeros when given to the base

10,

then

k+h

equals:

5

6

7

8

9

Answer: D

Concepts:factorial trailing zeros prime factorization number base

Difficulty rating: 2000

Small hint:

Count the powers of

2,

3,

and

5

in

15!

Big hint:

A base-

12

zero uses

2^2\cdot3

, while a base-

10

zero uses

2\cdot5

Solution:

The prime valuations are

\begin{gathered} v_2(15!)=11,\\ v_3(15!)=6,\\ v_5(15!)=3. \end{gathered}

Hence

k=\min(\lfloor\frac{11}{2}\rfloor,6)=5

in base

12,

and

h=\min(11,3)=3

in base

10.

Thus

k+h=8.

Therefore, the correct answer is D.