AMC 8 · 2013 · #1

Grade 4 number-theory

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

multiplesdivisibility-rulesmental-arithmetic systematic-enumeration ↑ Prerequisites: multi-digit-arithmeticmultiples

📏 Short solution 💡 2 insights

Problem

Danica wants to arrange her model cars in rows with exactly 6 cars in each row. She now has 23 model cars. What is the smallest number of additional cars she must buy in order to be able to arrange all her cars this way?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2013 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.

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Toolkit + CCSS Solution

Understand

Restated: Danica has $23$ model cars and wants to arrange them in rows of exactly $6$ each, with no empty seats and no leftovers. She needs to buy a few more cars to make this possible. What is the smallest number of additional cars she must buy?

Givens: Danica currently owns $23$ model cars; Each row must hold exactly $6$ cars; Every car must be in some row (no leftovers); Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $5$

Unknowns: The smallest number of additional cars Danica must buy

Understand

Givens: Danica currently owns $23$ model cars; Each row must hold exactly $6$ cars; Every car must be in some row (no leftovers); Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $5$

Plan

Primary tool: #2 Make a Systematic List

Secondary: #3 Eliminate Possibilities

Rows of exactly $6$ with no leftovers means the total must be a multiple of $6$. Tool #2 (Systematic List) is the natural move: list multiples of $6$ in order ($6, 12, 18, 24, \ldots$) and stop at the first one that is at least $23$. Tool #3 (Eliminate Possibilities) is a quick AMC sanity check — add each answer choice to $23$ and keep only the totals that are multiples of $6$, picking the smallest.

Execute — Answer: A

#2 Make a Systematic List 4.OA.B.4 Step 1

Recognize the rule: arranging $N$ cars into rows of exactly $6$ with no leftover means $N$ is a multiple of $6$.
So Danica's final total must be a multiple of $6$.

$N = 6 \times k$ for some whole number $k$

💡 Grouping into equal rows with nothing left over is the definition of "multiple of $6$" from Grade 4.

#2 Make a Systematic List 4.OA.B.4 Step 2

List the multiples of $6$ in order and stop at the first one that is at least $23$.

$$6, \; 12, \; 18, \; \boxed{24}, \; 30, \; \ldots$$

💡 Listing $6 \times 1, 6 \times 2, 6 \times 3, \ldots$ in order is exactly the Grade 4 "recognize multiples" skill.

#2 Make a Systematic List 1.OA.A.1 Step 3

The smallest multiple of $6$ that is at least $23$ is $24$.
So Danica's target total is $24$ cars.
Subtract her current $23$ to get the number of cars she still needs.

$$24 - 23 = 1$$

💡 Finding "how many more to reach the target" is a Grade 1 subtraction-within-$20$ word-problem move.

#3 Eliminate Possibilities 4.OA.B.4 Step 4

Verify against the answer choices by elimination.
For each choice $c$, check whether $23 + c$ is a multiple of $6$: $23+1=24$ ✓, $23+2=25$ ✗, $23+3=26$ ✗, $23+4=27$ ✗, $23+5=28$ ✗.
Only (A) works, so the answer is $\textbf{(A) } 1$.

$$23 + 1 = 24 = 6 \times 4 \;\Rightarrow\; \textbf{(A)}$$

💡 Plugging each choice into the multiple-of-$6$ test is the Grade 4 divisibility check applied as an AMC elimination.

[1] #2 4.OA.B.4 Recognize the rule: arranging $N$ cars into rows of exactly $6$ with no leftover

[2] #2 4.OA.B.4 List the multiples of $6$ in order and stop at the first one that is at least $2

[3] #2 1.OA.A.1 The smallest multiple of $6$ that is at least $23$ is $24$. So Danica's target t

[4] #3 4.OA.B.4 Verify against the answer choices by elimination. For each choice $c$, check whe

Review

Reasonableness: Sanity check: $23 \div 6 = 3$ remainder $5$. That remainder means Danica is $5$ cars into a fourth row of $6$, so she needs $6 - 5 = 1$ more car to complete that row. That matches the answer (A) and also matches the smallest-multiple-of-$6$-above-$23$ calculation, $24 - 23 = 1$.

Alternative: Tool #6 (Guess and Check) on whole numbers: start with $0$ extra cars ($23$, not a multiple of $6$ since $6 \times 3 = 18$ and $6 \times 4 = 24$), then $1$ extra ($24 = 6 \times 4$ ✓). Stop — $1$ works and nothing smaller does. Same answer (A).

CCSS standards used (min grade 4)

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite (Recognizing that "rows of exactly $6$ with no leftover" means the total must be a multiple of $6$, listing the multiples $6, 12, 18, 24, \ldots$, and using the divisibility-by-$6$ test to eliminate answer choices.)
1.OA.A.1 Solve addition and subtraction word problems within 20 (Computing the difference $24 - 23 = 1$ to find how many additional cars Danica needs to reach the target of $24$.)

⭐ This AMC 8 problem only needs the Grade 4 idea of "multiples" that you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 4)

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