2018 AMC 8 Problem 14
Attempt Problem 14 of the 2018 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
14.
Let be the greatest five-digit number whose digits have a product of What is the sum of the digits of
Answer: D
Small hint:
To maximize the five-digit number, maximize the digits from left to right.
Big hint:
Each chosen digit must divide the remaining required product.
Video solution:
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Written 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, D is the correct answer.
Problem 14 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
- 1994 AMC 8
- 1995 AMC 8
- 1996 AMC 8
- 1997 AMC 8
- 1998 AMC 8
- 1999 AMC 8
- 2000 AMC 8
- 2001 AMC 8
- 2002 AMC 8
- 2003 AMC 8
- 2004 AMC 8
- 2005 AMC 8
- 2006 AMC 8
- 2007 AMC 8
- 2008 AMC 8
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- 2012 AMC 8
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- 2014 AMC 8
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- 2025 AMC 8
- 2026 AMC 8