Competition · AMC preparation · step 4 of 4
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.
List the allowed primes
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 ListMultiply two primes for candidates
N 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 ListReject the perfect square
The 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 PossibilitiesTest the next candidate
The 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 ListConfirm nothing smaller works
Three 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.
The smallest number that is composite, is not a perfect square, and has every prime factor at least 50 must be the product of the two smallest different primes that are at least 50.
▸ Why?
Any qualifying number is composite, so it is a product of two or more prime factors, and condition (iv) forces every one of those primes to be at least 50.
▸ Why?
Every whole number greater than 1 breaks into a product of primes in exactly one way, and 'composite' means the number is not itself a single prime, so its prime factorization holds at least two prime factors.
▸ Why?
Among all such products, the smallest one uses as few prime factors as possible and picks the smallest allowed primes, because bringing in an extra prime factor or swapping in a larger prime only multiplies the value up.
▸ Why?
Multiplying by a larger amount, or multiplying in one more factor above one, piles more equal groups onto the total, so the product can only grow.
▸ Why?
Those two smallest primes must be two different primes, not one prime used twice, because a prime taken as a factor twice is that prime squared, and the conditions forbid a perfect square.
▸ Why?
Using the same prime as a factor two times makes two equal groups of that prime, which is exactly the prime multiplied by itself — a square.
This AMC 8 problem only needs Grade 6 prime-factorization reasoning: list the allowed primes, multiply the two smallest, and check the conditions.
- List the allowed primes
- Multiply two primes for candidates
- Reject the perfect square
- Test the next candidate
- Confirm nothing smaller works
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