Competition · AMC preparation · step 4 of 4
AMC 8 · 2013 · #1
Grade 4 number-theoryPick an answer.
AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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, …) 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.
Recognize the multiple of 6
Rows of exactly 6 with nothing left over means the final total N must be a multiple of 6.
Grouping into equal rows with nothing left over is the definition of "multiple of 6" from Grade 4.
For the cars to fill rows of exactly 6 with none left over, the total number of cars must be a multiple of 6.
▸ Why?
No car is left over means every car sits in some full row, and each full row holds exactly 6, so the cars split into a whole number of equal rows of 6.
▸ Why?
A whole number of equal rows of 6 is counted by adding 6 once for each row, which is 6 times the number of rows — and that is exactly what a multiple of 6 is.
List the multiples of 6
List the multiples of 6 in order and stop at the first one that is at least 23.
Listing 6 × 1, 6 × 2, 6 × 3, … in order is exactly the Grade 4 "recognize multiples" skill.
4.OA.B.4Make A Systematic ListFind the gap up to 24
The smallest multiple of 6 above 23 is 24, so the target is 24 cars; subtract the current 23 to get how many she still needs.
Finding "how many more to reach the target" is a Grade 1 subtraction-within-20 word-problem move.
1.OA.A.1Make A Systematic ListCheck against the choices
Add each choice to 23: only 23 + 1 = 24 lands on a multiple of 6, so the answer is (A).
Plugging each choice into the multiple-of-6 test is the Grade 4 divisibility check applied as an AMC elimination.
4.OA.B.4Eliminate PossibilitiesThis AMC 8 problem only needs the Grade 4 idea of "multiples" that you already know!
- Recognize the multiple of 6
- List the multiples of 6
- Find the gap up to 24
- Check against the choices
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