AMC 8 · 2006 · #10

Grade 5 geometry-2d

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

factorscoordinate-geometryarea-rectanglessystematic-enumeration systematic-enumeration ↑ Prerequisites: factorscoordinate-geometry

📏 Short solution 💡 2 insights 📊 Diagram

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Problem

Jorge must plot every ordered pair (w, l) of positive integers where w is the width and l is the length of a rectangle whose area is 12. Among five candidate graphs, choose the one whose dots are exactly those pairs.

Pick an answer.

(A)

Six points $(1,12), (2,6), (3,4), (4,3), (6,2), (12,1)$ on the curve $wl=12$

(B)

Six points on the diagonal line $l=w$: $(1,1), (3,3), (5,5), (7,7), (9,9), (11,11)$

(C)

Six points on the anti-diagonal line $w+l=12$: $(1,11), (3,9), (5,7), (7,5), (9,3), (11,1)$

(D)

Six horizontal points at height $l=6$: $(1,6), (3,6), (5,6), (7,6), (9,6), (11,6)$

(E)

Six vertical points at $w=6$: $(6,1), (6,3), (6,5), (6,7), (6,9), (6,11)$

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

The whole problem reduces to one question: which positive-integer pairs (w, l) have wl = 12? Tool #2 (Systematic List) handles this cleanly — sweep w = 1, 2, 3, … and keep only the values where l = 12/w is also a positive integer. Once the list is complete, Tool #1 (Draw a Diagram) translates each pair into a dot on the w-l plane and we match it to one of the five graphs. There is no need for algebra or special tricks; listing the factor pairs of 12 is the whole job.

1STEP 1

A rectangle's area is width × length, and here it is fixed at 12, so we need positive integers with wl = 12.

w × l = 12, w, l ∈ {1, 2, 3, …}

2STEP 2

Sweep w = 1, 2, 3, … keeping only divisors of 12; that leaves the six pairs (1,12), (2,6), (3,4), (4,3), (6,2), (12,1).

(1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1)

3STEP 3

Plot each pair as a dot with horizontal w and vertical l; the six dots fall on a curve sweeping down to the right.

Dots: (1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1)

4STEP 4

Only graph (A) shows exactly those six dots; (B),(C),(D),(E) use w=l, w+l=12, l=6, w=6 and miss wl=12.

Graph (A) matches → (A)

Answer

Six points (1,12), (2,6), (3,4), (4,3), (6,2), (12,1) on the curve wl=12

Quickly verify each dot in (A) by multiplying: 1 × 12 = 12, 2 × 6 = 12, 3 × 4 = 12, 4 × 3 = 12, 6 × 2 = 12, 12 × 1 = 12. All six products are 12, so every dot is a valid (w, l). The other graphs fail an immediate spot check: graph (B) includes (3, 3) with product 9 ≠ 12; graph (C) includes (1, 11) with product 11 ≠ 12; graph (D) includes (1, 6) with product 6 ≠ 12; graph (E) includes (6, 1) — that one works, but the rest of (E)'s dots like (6, 3) have product 18. Only (A) survives.

💡Key takeaway

Whenever a problem says "plot every (w, l) with width times length = N," it is really asking for the factor pairs of N. Sweep w = 1, 2, 3, …, keep the divisors, and you have all the dots.

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