Problem 27 · 2022 Math Kangaroo
Stretch
Number Theory
factorizationplace-value
The product of the digits of a number N is 20. Which of the following numbers cannot be the product of the digits of the number N + 1?
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Answer: D — 35
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Hint 1 of 2
A digit product of 20 forces specific digit sets like {4,5} or {1,4,5}.
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Hint 2 of 2
Compute N+1 for the small N with digit product 20 and see which target product never appears.
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Approach: enumerate N with digit product 20 and test N+1
- Numbers with digit product 20 use digits multiplying to 20 (e.g. 45, 54, 145, ...).
- For each, look at N+1 and its digit product; the reachable products include 24, 25, 30 and 40.
- A digit product of 35 = 5*7 would need a digit 7, which none of the N+1 cases produce, so 35 is impossible.
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