1997 AMC 8 Problem 23
Attempt Problem 23 of the 1997 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1997 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
23.
There are positive integers that have these properties:
I. the sum of the squares of their digits is and
II. each digit is larger than the one to its left.
The product of the digits of the largest integer with both properties is
Small hint:
Try the largest possible final digit and work downward
Big hint:
More digits make the minimum square-sum too large
Solution:
If the number had five digits, the smallest possible increasing positive digits would give square-sum already too large. So the largest valid number has at most four digits.
For a four-digit number with , the last digit cannot be or larger, since even .
Trying the remaining squares must sum to and This gives the valid number
Any number ending with or less is smaller than so the largest valid integer is
The product of its digits is
Thus, C is the correct answer.
Problem 23 in other years
- 1985 AMC 8
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