2018 AMC 8 Problems/Problem 14
Contents
Problem
Let be the greatest five-digit number whose digits have a product of . What is the sum of the digits of ?
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 (Factorial)
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:
.
Video Solution (CREATIVE ANALYSIS!!!)
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Video Solutions
https://youtu.be/7an5wU9Q5hk?t=13
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See Also
2018 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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All AJHSME/AMC 8 Problems and Solutions |
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