Problem:
Walter rolls four standard six-sided dice and finds that the product of the numbers on the upper faces is . Which of the following could not be the sum of the upper four faces?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Factor into primes, , and notice that there are at most two 's and no 's among the numbers rolled. If there are no 's, then there must be two 's since these are the only values that can contribute to the prime factorization. In this case, the four 's in the factorization must be the result of two 's in the roll. Hence the sum is a possible value for the sum. Next, consider the case with just one . Then there must be one , and the three remaining 's must be the result of a and a . Thus, the sum is also possible. Finally, if there are two 's, then there must also be two 's or a and a , with sums of and . Hence is the only sum not possible.
Since does not divide and , there can be no 's and at most two 's. Thus, the only ways the four dice can have a sum of are: ; ; and . Since none of these products is , the answer is .