1998 AMC 12 Problem 13

Attempt Problem 13 of the 1998 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 1998 AMC 12 solutions, or check the answer key.

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13.

Walter rolls four standard six-sided dice and finds that the product of the numbers on the upper faces is

144.

Which of the following could not be the sum of the upper four faces?

14

15

16

17

18

Factor

144=2^4\cdot3^2

using four factors from

1

through

6

Separate the cases with zero, one, or two faces showing

6

Valid rolls include

(4,4,3,3)

with sum

14,

(6,4,3,2)

with sum

15,

(6,6,2,2)

with sum

16,

and

(6,6,4,1)

with sum

17.

If two

6

s occur, the other two faces have product

4

and sum at most

5,

giving total at most

17;

the cases with fewer

6

s also cannot total

18

while retaining product

144.

Thus

18

is impossible, and E is correct.

Problem 13 in other years