Problem 3 · 1993 AJHSME
Medium
Number Theory
prime-factorization
Which of the following numbers has the largest prime factor?
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Answer: B — 51.
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Hint 1 of 2
The question isn't 'which number is biggest' — it's 'which has the biggest prime piece.' A small-looking number can hide a large prime. Break each into its prime building blocks.
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Hint 2 of 2
Each of these is a product of just two primes. Find the larger prime in each pair, then compare those — that's the only number that matters per choice.
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Approach: factor each, then race the largest primes
- Split each into primes: 39 = 3·13, 51 = 3·17, 77 = 7·11, 91 = 7·13, 121 = 11·11. The biggest prime inside each is 13, 17, 11, 13, 11.
- The champion is 17, living inside 51 — even though 51 is far from the largest number on the list.
- Why this transfers: 'largest prime factor' is about the deepest prime in the factor tree, not the size of the number. The two ideas come apart — that's exactly the surprise these problems test.
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