AMC 8 · 2013 · #1
Easy mode Grade 4Problem
Danica is lining up her model cars in rows. She wants exactly cars in every row, with no empty spots and no row left short.
Right now she has 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.
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!