Problem 22 · 1995 AJHSME
Hard
Number Theory
factoring
The number 6545 can be written as a product of a pair of positive two-digit numbers. What is the sum of this pair of numbers?
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Answer: A — 162.
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Hint 1 of 2
You can't see the two-digit factors by staring at 6545 — but its prime factorization reveals every possible way to split it. Break it into primes first; the answer is just a regrouping of those.
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Hint 2 of 2
6545 ends in 5, so 5 comes out immediately. Keep pulling small primes, then sort the prime pieces into two groups that each land between 10 and 99.
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Approach: prime-factor, then regroup the primes into two two-digit numbers
- Break 6545 down: it ends in 5 → divide by 5 to get 1309; 1309 = 7 × 187 = 7 × 11 × 17. So 6545 = 5 × 7 × 11 × 17.
- Now bundle those four primes into two two-digit numbers. The bundle (5 × 17) × (7 × 11) = 85 × 77 works — both are two-digit. (5×7=35 with 11×17=187 fails, since 187 is three digits.)
- Sum = 85 + 77 = 162.
- Why this transfers: the prime factorization is the master list of a number's building blocks — every factor pair is just one way of dividing those primes into two teams. Factor first, then the regrouping is quick.
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