Problem 11 · 2021 Math Kangaroo
Hard
Number Theory
factorizationdivisibility
What fraction of all the divisors of \(7!\) is odd?
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Answer: D — \(\tfrac{1}{5}\)
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Hint 1 of 2
Write 7! as a product of prime powers.
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Hint 2 of 2
Odd divisors come only from the odd primes; compare their count to all divisors.
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Approach: count divisors via prime factorisation
- 7! = 5040 = 2⁴ · 3² · 5 · 7, with (4+1)(2+1)(1+1)(1+1) = 60 divisors.
- Odd divisors drop the factor 2: from 3² · 5 · 7 there are 3 · 2 · 2 = 12.
- Fraction odd = 12/60 = 1/5.
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