2018 AMC 8
Complete problem set with solutions and individual problem pages
Let be the greatest five-digit number whose digits have a product of . What is the sum of the digits of ?
- A.
- B.
- C.
- D.
- E.
Solution 1
If we start off with the first digit, we know that it can't be since is not a factor of . We go down to the digit , which does work since it is a factor of . Now, we have to know what digits will take up the remaining four spots. To find this result, just divide . The next place can be , as it is the largest factor, aside from . Consequently, our next three values will be and if we use the same logic. Therefore, our five-digit number is , so the sum is .
Solution 2
is , so we have . (Alternatively, you could identify the prime factors .) Now look for the largest digit you can create by combining these factors.
Use this largest digit for the ten-thousands place: _ , _ _ _
Next you use the and the for the next places: _ _ (You can't use because the was used to make .)
Fill the remaining places with 1:
.
