Problem 25 · 2022 Math Kangaroo
Stretch
Number Theory
casework
How many three-digit numbers are there that are equal to five times the product of their digits?
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Answer: A — 1
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Hint 1 of 2
You need a three-digit number equal to five times the product of its digits.
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Hint 2 of 2
The number must be a multiple of 5, so its last digit is 0 or 5; a 0 kills the product, so it ends in 5.
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Approach: search constrained three-digit numbers
- Since n = 5·(product of digits), n is a multiple of 5; the units digit can't be 0, so it is 5.
- Checking the candidates, only 175 works: 5·1·7·5 = 175.
- So there is exactly 1 such number.
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