AMC 8 · 2003 · #2

Grade 4 number-theory

Archetype Small-Domain Constrained Search

prime-numbersprime-factorizationparitydivisibility-rules systematic-enumerationcasework ↑ Prerequisites: primality-testmultiples

📏 Short solution 💡 2 insights

📘 View easy version →

Problem

Which of the following numbers has the smallest prime factor?

$\mathrm{(A)}\ 55 \qquad\mathrm{(B)}\ 57 \qquad\mathrm{(C)}\ 58 \qquad\mathrm{(D)}\ 59 \qquad\mathrm{(E)}\ 61$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

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

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

#5 Look for a Pattern 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 $2$.

$$\text{Primes in order: } 2, 3, 5, 7, 11, \ldots$$

💡 Knowing the order of primes turns the problem into a quick checklist: try $2$ first, then $3$, then $5$.

#3 Eliminate Possibilities 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.

$$55, 57, 59, 61 \text{ are odd}; \quad 58 \text{ is even}$$

💡 The even/odd pattern is the fastest divisibility test there is — one glance at the ones digit decides it.

#3 Eliminate Possibilities 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.

$$58 = 2 \times 29 \;\Rightarrow\; \text{smallest prime factor of } 58 = 2 \;\Rightarrow\; \textbf{(C)}$$

💡 Once one candidate hits the smallest possible prime, the search is over — no later check can produce a smaller answer.

[1] #5 4.OA.B.4 List the primes from smallest to largest. The smallest prime factor any number c

[2] #3 3.OA.D.9 Check each candidate for divisibility by $2$. A number is divisible by $2$ exact

[3] #3 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.4 Find 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.9 Identify 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).

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 4)

More like this