AMC 8 · 2013 · #1
Easy mode Grade 4Problem
Danica is lining up her model cars in rows. She wants exactly 6 cars in every row, with no empty spots and no row left short.
Right now she has 23 cars. She plans to buy a few more so it works out evenly.
What is the smallest number of extra cars she needs to buy?
Pick an answer.
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.
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
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) $5$
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
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 $6$.
💡 Grouping into equal rows with nothing left over is the definition of "multiple of $6$" from Grade 4.
4.OA.B.4 Step 2 List the multiples of $6$ in order and stop at the first one that is at least $23$.
💡 Listing $6 \times 1, 6 \times 2, 6 \times 3, \ldots$ in order is exactly the Grade 4 "recognize multiples" skill.
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.
💡 Finding "how many more to reach the target" is a Grade 1 subtraction-within-$20$ word-problem move.
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 $\textbf{(A) } 1$.
💡 Plugging each choice into the multiple-of-$6$ test is the Grade 4 divisibility check applied as an AMC elimination.
4.OA.B.4 Recognize the rule: arranging $N$ cars into rows of exactly $6$ with no leftover 4.OA.B.4 List the multiples of $6$ in order and stop at the first one that is at least $2 1.OA.A.1 The smallest multiple of $6$ that is at least $23$ is $24$. So Danica's target t 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.4Find 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.1Solve 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!
⭐ This AMC 8 problem only needs the Grade 4 idea of "multiples" that you already know!
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