2004 AMC 10B Problem 4

Question

A standard six-sided die is rolled, and

P

is the product of the five numbers that are visible. What is the largest number that is certain to divide

P?

6

12

24

144

720

The product of all six faces is

6! = 720

; exactly one face is hidden

For each prime, find how many copies must remain no matter which face is hidden

Since

6! = 720 = 2^4 \cdot 3^2 \cdot 5,

the visible product uses only the primes

2,

3,

and

5.

Hiding

4

leaves the fewest

2

’s, namely

2^2.

Hiding

3

or

6

leaves the fewest

3

’s, namely one. Hiding

5

leaves no factor of

5.

Therefore

P

is always divisible by

2^2 \cdot 3 = 12,

but not necessarily by any larger number.

Thus, the correct answer is B.

Problem 4 in other years