Problem 13 · 2012 AMC 8
Medium
Number Theory
gcd
Jamar bought some pencils costing more than a penny each at the school bookstore and paid $1.43. Sharona bought some of the same pencils and paid $1.87. How many more pencils did Sharona buy than Jamar?
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Answer: C — 4 more pencils.
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Hint 1 of 2
Switch to whole cents (143 and 187) so you're working with integers. A whole number of pencils at a whole-cent price means that price divides each total exactly — so the price is a common factor of 143 and 187.
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Hint 2 of 2
This is a shared divisor hunt: the price divides both amounts, so it divides their gcd. Factor each amount and the price has to fall out (the "more than a penny" rules out the trivial 1¢).
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Approach: the price is a common factor of both totals
- Some whole number of pencils times the (whole-cent) price gives 143¢, and likewise 187¢. So the price divides both 143 and 187 — it's a common factor.
- Factor them: 143 = 11 × 13 and 187 = 11 × 17. Their only shared factor above 1 is 11, and "more than a penny" forbids 1¢, so each pencil costs 11¢.
- The clean finish: Sharona paid 187 − 143 = 44¢ more, which is 44 / 11 = 4 extra pencils — no need to find each kid's count.
- The transferable idea: "same whole-number price buys both totals" means the price is a common divisor; factoring or gcd pins it down. Subtracting the totals before dividing skips the individual counts.
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