AMC 8 · 2001 · #2
Easy mode Grade 4Problem
I am thinking of two whole numbers.
When you multiply them, you get 24. When you add them, you get 11.
What is the larger of the two numbers?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: Two whole numbers have product $24$ and sum $11$. Find the larger of the two numbers.
Givens: The two numbers are whole numbers (non-negative integers); Their product is $24$; Their sum is $11$; Answer choices: (A) $3$, (B) $4$, (C) $6$, (D) $8$, (E) $12$
Unknowns: The larger of the two whole numbers
Understand
Restated: Two whole numbers have product $24$ and sum $11$. Find the larger of the two numbers.
Givens: The two numbers are whole numbers (non-negative integers); Their product is $24$; Their sum is $11$; Answer choices: (A) $3$, (B) $4$, (C) $6$, (D) $8$, (E) $12$
Plan
Primary tool: #2 Make a List
Secondary: #6 Guess and Check
Because the product is $24$, the two numbers must be a factor pair of $24$. There are only four positive whole-number factor pairs, so Tool #2 (Make a List) lets us write them all down in a few seconds. For each pair, Tool #6 (Guess and Check) does the rest: just add the two numbers and see whether the sum is $11$. No algebra is needed — the short list does all the work.
Execute — Answer: D
4.OA.B.4 Step 1 - List all positive whole-number factor pairs of $24$.
- Start with $1$ and walk up through the divisors of $24$.
💡 Grade 4 factor-pair work: list divisors in order so no pair is missed.
4.OA.A.3 Step 2 Add the two numbers in each pair and compare each sum to $11$.
💡 Only one row meets both conditions, so that pair must be the answer.
4.OA.A.3 Step 3 - The matching pair is $3$ and $8$.
- The question asks for the larger number.
💡 Once the right pair is found, picking the larger value is a final reading-the-question check.
4.OA.B.4 List all positive whole-number factor pairs of $24$. Start with $1$ and walk up 4.OA.A.3 Add the two numbers in each pair and compare each sum to $11$. 4.OA.A.3 The matching pair is $3$ and $8$. The question asks for the larger number. Review
Reasonableness: Verify both conditions on the chosen pair: $3 \times 8 = 24$ matches the product, and $3 + 8 = 11$ matches the sum. Both hold, so $8$ is correct. Choice (E) $12$ can be ruled out quickly because its partner under product $24$ is $2$, and $2 + 12 = 14 \ne 11$. Choices (A) $3$ and (D) $8$ name the two members of the right pair; the question asks for the larger, so (D) wins over (A).
Alternative: Tool #6 (Guess and Check) on the answer choices: try each choice as the larger number and see whether the partner forced by sum $11$ multiplies to $24$. (A) $3$: partner $8$, $3 \times 8 = 24$ — works, but $3$ is the smaller. (D) $8$: partner $3$, same pair, and $8$ is the larger. Same answer (D), reached by checking the choices instead of listing factor pairs.
CCSS standards used (min grade 4)
4.OA.B.4Find all factor pairs for a whole number in the range 1-100; recognize that a whole number is a multiple of each of its factors (Listing the four factor pairs of $24$: $(1,24), (2,12), (3,8), (4,6)$.)4.OA.A.3Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations (Adding the two numbers in each factor pair, comparing each sum to $11$, and then selecting the larger member of the winning pair.)
⭐ When a problem gives both the product and the sum of two whole numbers, list the factor pairs of the product first — there are usually only a handful — and pick the one whose sum matches. Here the factor pair $(3, 8)$ adds to $11$, so the larger number is $8$, choice (D).
⭐ When a problem gives both the product and the sum of two whole numbers, list the factor pairs of the product first — there are usually only a handful — and pick the one whose sum matches. Here the factor pair $(3, 8)$ adds to $11$, so the larger number is $8$, choice (D).