Problem 1 · 2013 AMC 8
Easy
Number Theory
next-multiple
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?
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Answer: A — 1 more car.
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Hint 1 of 2
"Exactly 6 per row" is a disguised way of saying "the total must be a multiple of 6." So really: what's the next multiple of 6 once you pass 23?
Still stuck? Show hint 2 →
Hint 2 of 2
When a word problem demands a whole number of equal groups, translate it to "the total is a multiple of (group size)" and round up to the nearest one.
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Approach: round up to the next multiple of 6
- Rows of exactly 6 means the total must be a multiple of 6 — that's the hidden condition. So 23 itself won't work.
- Count up from 23 to the next multiple of 6: 24 = 6 × 4. She needs 24 − 23 = 1 more car.
- You'll see this again: any "arrange in equal rows / groups" problem is secretly asking for a multiple, so the answer is always (next multiple) − (what you have).
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