AMC 8 · 2003 · #2
Grade 4 number-theoryProblem
Which of the following numbers has the smallest prime factor?
Pick an answer.
AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: From the five numbers $55, 57, 58, 59, 61$, find the one whose smallest prime factor is the smallest. In other words, look at each number's smallest prime divisor and pick whichever number has the smallest such divisor.
Givens: Five candidate numbers: $55, 57, 58, 59, 61$; We compare them by their smallest prime factor, not by their size; Answer choices: (A) $55$, (B) $57$, (C) $58$, (D) $59$, (E) $61$
Unknowns: Which of the five numbers has the smallest prime factor
Understand
Restated: From the five numbers $55, 57, 58, 59, 61$, find the one whose smallest prime factor is the smallest. In other words, look at each number's smallest prime divisor and pick whichever number has the smallest such divisor.
Givens: Five candidate numbers: $55, 57, 58, 59, 61$; We compare them by their smallest prime factor, not by their size; Answer choices: (A) $55$, (B) $57$, (C) $58$, (D) $59$, (E) $61$
Plan
Primary tool: #3 Eliminate Possibilities
Secondary: #5 Look for a Pattern
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 $2$.
Execute — Answer: C
4.OA.B.4 Step 1 - List the primes from smallest to largest.
- The smallest prime factor any number can possibly have is $2$.
- After that comes $3$, then $5$, and so on.
- So the search for "smallest prime factor" starts at $2$.
💡 Knowing the order of primes turns the problem into a quick checklist: try $2$ first, then $3$, then $5$.
3.OA.D.9 Step 2 - Check each candidate for divisibility by $2$.
- A number is divisible by $2$ exactly when it is even (ones digit is $0, 2, 4, 6,$ or $8$).
- Scan the ones digits: $55, 57, 58, 59, 61$ end in $5, 7, 8, 9, 1$.
- 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.
4.OA.B.4 Step 3 - Conclude with the smallest-prime rule.
- Since $58$ is divisible by $2$, its smallest prime factor is $2$.
- The other four numbers are odd, so their smallest prime factor is at least $3$.
- Nothing can beat $2$, so $58$ wins.
💡 Once one candidate hits the smallest possible prime, the search is over — no later check can produce a smaller answer.
4.OA.B.4 List the primes from smallest to largest. The smallest prime factor any number c 3.OA.D.9 Check each candidate for divisibility by $2$. A number is divisible by $2$ exact 4.OA.B.4 Conclude with the smallest-prime rule. Since $58$ is divisible by $2$, its small Review
Reasonableness: Confirm by finding the smallest prime factor of each candidate. $55 = 5 \times 11$, smallest prime $5$. $57 = 3 \times 19$, smallest prime $3$. $58 = 2 \times 29$, smallest prime $2$. $59$ is prime, smallest prime $59$. $61$ is prime, smallest prime $61$. The smallest of $\{5, 3, 2, 59, 61\}$ is $2$, achieved by $58$. The answer (C) holds.
Alternative: Tool #2 (Make a Systematic List): for each number $55, 57, 58, 59, 61$, list factor pairs and read off the smallest prime in each. $55: 5 \cdot 11$; $57: 3 \cdot 19$; $58: 2 \cdot 29$; $59: \text{prime}$; $61: \text{prime}$. Compare the smallest primes: $5, 3, 2, 59, 61$. The minimum is $2$, from $58$, giving (C).
CCSS standards used (min grade 4)
4.OA.B.4Find all factor pairs for a whole number in the range 1-100; determine whether a given whole number is prime or composite (Knowing what "prime factor" means, and that the primes in order start $2, 3, 5, \ldots$, so the search for a smallest prime factor begins at $2$.)3.OA.D.9Identify arithmetic patterns, including patterns in the addition table or multiplication table, and explain them using properties of operations (Using the even/odd pattern (divisibility by $2$) to scan the candidates at a glance and find the one with $2$ as a factor.)
⭐ 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).
⭐ 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).
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