Problem 28 · 2021 Math Kangaroo
Stretch
Counting & Probability
factorizationcareful-counting
How many five-digit positive numbers have the product of their digits equal to 1000?
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Answer: D — 40
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Hint 1 of 2
1000 = 2³×5³; only the digit 5 can supply a factor of 5.
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Hint 2 of 2
So three digits are 5; the remaining two multiply to 8 — list those digit-pairs and count arrangements.
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Approach: fix the three 5s, then place the factor-8 pair
- 1000 = 2³×5³, and only the digit 5 contributes a factor 5, so exactly three digits are 5.
- The other two digits multiply to 8: the multisets are {1,8} and {2,4}.
- Each five-digit multiset {5,5,5,x,y} has 5!/3! = 20 arrangements.
- Two such multisets give 20+20 = 40 numbers, choice (D).
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