Competition · AMC preparation · step 4 of 4
AMC 8 · 2003 · #2
Grade 4 number-theoryPick an answer.
AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
This is a multiple-choice problem with only five candidates, so Tool #3 (Eliminate Possibilities) is the natural fit: test each choice against the smallest primes in order. Tool #5 (Look for a Pattern) sharpens the test — the smallest prime is 2, and the pattern "divisible by 2 = even" lets us scan the list at a glance. If any candidate is even, it must be the winner, because no number can have a prime factor below 2.
List the primes in order
List the primes in order: the smallest one any number can have is 2, then 3, then 5 — so begin the hunt at the bottom, with 2.
Knowing the order of primes turns the problem into a quick checklist: try 2 first, then 3, then 5.
4.OA.B.4Look For A PatternTest each choice for evenness
A number is divisible by 2 exactly when it is even, so scan the ones digits (5, 7, 8, 9, 1): only 58 ends in an even digit.
The even/odd pattern is the fastest divisibility test there is — one glance at the ones digit decides it.
3.OA.D.9Eliminate PossibilitiesName the smallest prime factor
58 is even, so its smallest prime factor is 2; the other four are odd (smallest prime at least 3), and nothing beats 2 — so (C).
Once one candidate hits the smallest possible prime, the search is over — no later check can produce a smaller answer.
An even number's smallest prime factor is 2, and 2 is the smallest prime factor any whole number can have.
▸ Why?
An even number is divisible by 2, and it is built out of prime numbers, so 2 is one of its prime factors; since no prime is smaller than 2, that 2 is the number's smallest prime factor — and because 2 is the smallest prime of all, no whole number can have a prime factor, and hence no smallest prime factor, below 2.
▸ Why?
An even number splits into two equal whole groups, which is exactly what it means for 2 to divide it with nothing left over.
▸ Why?
Every whole number above 1 is a product of primes in exactly one way, so the number is built from a definite list of primes; the 2 that divides an even number is one of those building-block primes, so it genuinely sits among the number's prime factors.
The smallest prime is 2, so on a "smallest prime factor" question, first ask: is any choice even? If yes, that choice wins immediately — here 58 is the only even number, so the answer is (C).
- List the primes in order
- Test each choice for evenness
- Name the smallest prime factor
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