AMC 10 · 2012 · #10

Grade 7 algebra

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

divisor-countprime-factorizationfactors systematic-enumeration ↑ Prerequisites: prime-factorization

📏 Short solution 💡 2 insights

Problem

The proportion

\frac{M}{6}=\frac{6}{N}

must hold, where M and N are positive integers. Because the pair is written in order, (M,N) and (N,M) count as different unless the two entries happen to be equal. Find how many ordered pairs (M,N) satisfy the proportion.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

The proportion hides a simpler equation, so Tool #4 (Introduce a Variable) is used first to cross-multiply M/6=6/N into MN=36. Once the problem becomes "how many positive-integer factor pairs give 36," Tool #2 (Make a Systematic List) walks the divisors of 36 in order so none are missed or double-counted. Tool #3 (Eliminate Possibilities) guards against the classic trap of counting unordered pairs (which would give 5) instead of ordered ones.

1STEP 1

Cross-multiply the proportion

Two equal fractions cross-multiply, so $\frac{M}{6}=\frac{6}{N}$ becomes MN=36.

M/6=6/N → MN=6 × 6=36

2STEP 2

See it as factor pairs of 36

For each divisor M of 36 the partner N= $\frac{36}{M}$ is forced, so pairs and divisors match one for one.

N=36/M, valid exactly when M ∣ 36

3STEP 3

List the divisors of 36 in order

The divisors 1,2,3,4,6,9,12,18,36 give 9 ordered pairs, with (6,6) its own reverse — choice (D).

1,2,3,4,6,9,12,18,36 → 9 pairs → (D)

Answer

The count of ordered pairs equals the number of divisors of 36. Using prime factorization, 36=2² · 3², so the divisor count is (2+1)(2+1)=9 — matching the hand list. If you mistakenly counted unordered pairs you would get 5 (the four flip-pairs plus (6,6)), which is not even an answer choice, confirming that ordered counting is intended and 9 is right.

💡Key takeaway

Cross-multiply the proportion to get MN=36, then count the divisors of 36 — each divisor is one ordered pair.

Cross-multiply the proportion
See it as factor pairs of 36
List the divisors of 36 in order

Cross-multiply the proportion

See it as factor pairs of 36

List the divisors of 36 in order

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