AMC 8 · 2012 · #13

Grade 6 number-theory

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Multiplicative Rate or Series

gcdprime-factorizationdivisibility-rulesfactors convert-to-algebraidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticprime-factorization

📏 Medium solution 💡 3 insights

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Problem

At the same school bookstore, Jamar paid

1.43</span> for some pencils and Sharona paid <span class="mk-c">

1.87 for some of the same pencils. Each pencil costs a whole number of cents that is more than 1. How many more pencils did Sharona buy than Jamar?

Pick an answer.

(A)

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(B)

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(C)

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(D)

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(E)

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AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Guess and Check

The price per pencil c has to divide both 143 and 187, and the problem says c ＞ 1. With Tool #13 (Convert to Algebra) we write the two simple equations j · c = 143 and s · c = 187, which makes "c is a common divisor" obvious. Then Tool #6 (Guess and Check) is the lightest way to find c — try small primes (2, 3, 5, 7, 11, …) on 143 until one fits, and confirm it also divides 187. Tool #7 (Identify Subproblems) splits the work into three clean steps: find c, then find j and s, then subtract.

1STEP 1

Turn both totals into cents and name the unknowns c, j, s, so the equations are j·c = 143 and s·c = 187 with c ＞ 1.

j · c = 143, s · c = 187, c ＞ 1

2STEP 2

Since c divides both totals, test small primes on 143 and find 143 = 11 × 13.

143 ÷ 2 = 71.5 no; 143 ÷ 3: digit sum 8, no; 143 ÷ 5: doesn't end in 0 or 5, no; 143 ÷ 7 = 20.4… no; 143 ÷ 11 = 13 yes → 143 = 11 × 13

3STEP 3

Check the same prime on the other total: 187 = 11 × 17, so 11 divides both.

187 ÷ 11 = 17 → 187 = 11 × 17

4STEP 4

The only common divisors of 143 and 187 are 1 and 11, and c ＞ 1 rules out 1, so c = 11 cents is forced.

gcd(143, 187) = 11

5STEP 5

Divide back: j = 13 and s = 17, so Sharona bought s - j = 4 more pencils — (C).

j = 143 ÷ 11 = 13, s = 187 ÷ 11 = 17 → s - j = 17 - 13 = 4 → (C)

Answer

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Sanity check the units and the size of the answer. At 11 cents each, 13 pencils cost 13 × 11 = 143 cents =

1.43 and 17 pencils cost 17 × 11 = 187 cents =

1.87 — both totals match exactly. The difference 17 - 13 = 4 is a small positive integer, matching one of the answer choices. If we had picked c = 1 (a penny), the count difference would be 187 - 143 = 44, which isn't an option — confirming the problem's c ＞ 1 hint really is the key constraint.

💡Key takeaway

This AMC 8 problem just needs Grade 6 GCF thinking: find the only price bigger than 1 cent that divides both totals, then divide to get the counts.

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