Problem:
Let be the greatest five-digit number whose digits have a product of . What is the sum of the digits of
Answer Choices:
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Solution:
To make the largest possible -digit number, we must maximize the first digit (the digit in the ten-thousands place). The largest number that is strictly less than and divides is , so the first digit must be . Therefore, the product of the remaining number is . Similarly, we must now maximize the second digit. The largest number that is less than and divides is , so the second digit is . Therefore, the product of the remaining number is . We must then maximize the third digit. The largest number that is less than and divides is , so the third digit is . Therefore, the product of the remaining number is . This means the th and th digits are . This makes , so the sum of the digits is
Thus, is the correct answer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions