Problem 23 · 2025 Math Kangaroo
Hard
Number Theory
primesfactorization
The product of three prime numbers is 11 times their sum. What is the maximum sum of the three numbers?
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Answer: E — 26
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Hint 1 of 2
11 times the sum being a product of three primes strongly suggests 11 is one of them.
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Hint 2 of 2
With one prime = 11, reduce to (p−1)(q−1) = 12 and maximize p+q.
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Approach: force 11 as a factor, then factor the rest
- Let the primes be 11, p, q. Then 11pq = 11(11+p+q), so pq = 11+p+q.
- Rearranged: (p−1)(q−1) = 12. To maximize the sum take p−1=12, q−1=1 → p=13, q=2 (both prime).
- Sum = 11+13+2 = 26.
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