AMC 8 · 2012 · #18
Grade 6 number-theoryPick an answer.
AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The phrase "smallest positive integer with no prime factor less than 50" is a classic Tool #2 setup: list the allowed primes (53, 59, 61, …) in order, then list the candidate products in order of size. Because N must be composite, the smallest products to try are 53 × 53, 53 × 59, 53 × 61, … — easy to walk through in order. Tool #3 (Eliminate) then closes the deal: the problem is multiple choice, so once we find a valid candidate that matches a choice, the smaller-or-equal choices can be checked and eliminated.
Condition (iv) forces every prime factor to be at least 50, so the usable primes start at 53, 59, 61, 67 (51 and 57 aren't prime).
Recognizing primes versus composites in the 50s is exactly the Grade 4 "factors and primes" skill.
4.OA.B.4Make A Systematic ListN is composite, so multiply the smallest allowed primes in order — the candidates to test are 53 × 53, 53 × 59, 53 × 61, …
Building composite numbers from their smallest prime factors is the prime-factorization reasoning of Grade 6 number sense.
6.NS.B.4Make A Systematic ListThe first candidate 53 × 53 = 2809 is a perfect square, so condition (iii) eliminates it despite its big prime factor.
Spotting p × p = p² as a perfect square uses the Grade 6 exponent definition.
6.EE.A.1Eliminate PossibilitiesThe next candidate 53 × 59 gives 3127: positive, composite, factors 53 and 59 distinct (not a square), both at least 50 — all four hold.
Reading the prime factorization 53¹ · 59¹ to confirm "composite but not a square" is core Grade 6 factor reasoning.
6.NS.B.4Make A Systematic ListThree primes give at least 53³ = 148,877, and the only smaller two-prime product is the banned 53 × 53, so 3127 is smallest — choice (A).
Comparing factorizations to rule out smaller composites uses Grade 6 factor and multiple reasoning.
6.NS.B.4Eliminate PossibilitiesThis AMC 8 problem only needs Grade 6 prime-factorization reasoning: list the allowed primes, multiply the two smallest, and check the conditions.