AMC 8 · 2000 · #7

Grade 7 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

multi-digit-arithmeticsystematic-enumerationparity caseworksystematic-enumerationoptimization-counting ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 3 insights

Problem

From the set {-8, -6, -4, 0, 3, 5, 7}, pick three different numbers and multiply them. What is the smallest (most negative) product you can make?

Pick an answer.

(A)

-336

(B)

-280

(C)

-210

(D)

-192

(E)

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Extreme Principle

We want the most negative product, so Tool #14 (Extreme Principle) applies: the answer comes from pushing the chosen numbers to extreme magnitudes with the right signs. The sign rules narrow the search: three numbers multiply to a negative only when the count of negatives is 1 or 3, so Tool #2 (List Out Cases) gives just two cases to compare. Inside each case, the Extreme Principle picks the numbers with the largest absolute values. Skip 0 — any product containing it is 0, which is not the smallest.

1STEP 1

The product is negative only when an odd number of factors are negative — 1 or 3 negatives; any pick with 0 gives 0, so drop 0.

negative product ⇔ #negatives ∈ {1, 3}

2STEP 2

Case A takes all three negatives -8, -6, -4: two negatives make a positive, the third flips it back to -192.

(-8) × (-6) × (-4) = 48 × (-4) = -192

3STEP 3

Case B (one negative, two positives): the Extreme Principle takes the biggest-size negative and two biggest positives — -8, 7, 5.

|-8| ＞ |-6| ＞ |-4| and 7 ＞ 5 ＞ 3 → pick -8, 7, 5

4STEP 4

Case B gives (-8)×7×5 = -280; since -280 ＜ -192 on the number line, the minimum is -280, choice (B).

(-8) × 7 × 5 = -280; -280 ＜ -192 → minimum = -280 → (B)

Answer

-280

Spot-check by trying other Case B picks. Using -6 instead of -8: (-6) × 7 × 5 = -210 (choice C) — less negative than -280, so -8 was the right pick. Using -8 but with 7 and 3: (-8) × 7 × 3 = -168 — less negative. Choice (A) -336 would need |-8| × |two numbers| = 336, i.e. two positives whose product is 42 — but the largest available pair gives 7 × 5 = 35, not 42, so -336 is unreachable. The answer (B) -280 holds.

💡Key takeaway

For a smallest-product question, list only the sign patterns that go negative, then within each one make every factor as big in size as possible. Two short cases beat any guess-and-shuffle approach.

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