Problem 18 · 2012 AMC 8
Medium
Number Theory
product-of-distinct-primes
What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?
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Answer: A — 3127.
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Hint 1 of 2
Don't search numbers — decode the three conditions into a recipe for what the number must be built from. Translate each phrase: "not prime", "no prime factor < 50", "not a square."
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Hint 2 of 2
"Not prime" + "every prime factor ≥ 50" ⇒ it's a product of at least two primes, all ≥ 50. "Not a square" ⇒ those primes must be different. So: smallest two different primes above 50.
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Approach: translate each condition, then take the smallest legal build
- Read the conditions as a blueprint. "Not prime" means it factors into ≥ 2 primes; "no prime factor below 50" means every one of those primes is ≥ 50; "not a square" means we can't repeat a prime (like p²). So the number = a product of distinct primes ≥ 50.
- To make it smallest, use the two smallest primes above 50. Checking 51 = 3×17 (no), 52, 53 is prime; next prime is 59.
- Smallest such product: 53 × 59 = 3127.
- The skill: turn each word-condition into a structural fact about the prime factorization first — then "smallest" just means "use the smallest allowed ingredients."
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