Problem 18 · 2023 Math Kangaroo
Stretch
Number Theory
factorizationdigit-sum
For each positive integer n the number \(n!\) is defined as the product of all numbers from 1 to n. For example, \(4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24\). For a certain N the formula \(N! = 6! \cdot 7!\) holds. How big is the sum of the digits of N?
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Answer: A — 1
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Hint 1 of 2
Try to write 6!·7! as a single factorial.
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Hint 2 of 2
Notice 7! = 7·6!, and look for a factorial equal to 6!·7!.
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Approach: recognise 6!·7! as 10!
- Since 7! = 7·6!, the product 6!·7! equals 3 628 800.
- That value is exactly 10!, so N = 10.
- The digit sum of 10 is 1 + 0 = 1.
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