AMC 10 · 2013 · #5

Grade 7 algebra
Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In
optimizationfactors extremal-construction ↑ Prerequisites: factors
📏 Medium solution 💡 2 insights
Problem
Two positive integers a and b are each less than 6, so each is one of 1, 2, 3, 4, 5. Find the smallest possible value of 2·a - a·b.

Pick an answer.

(A)
$\ -20$
(B)
$\ -15$
(C)
$\ -10$
(D)
$\ 0$
(E)
$\ 2$

AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Extreme Principle

The question asks for a minimum over a small, bounded set, which is the signature of Tool #14 (Extreme Principle): the smallest value lives at a boundary, so test the edge values rather than all 25 pairs. First rewrite the expression by factoring (Tool #4, treating the two letters as one product) so the sign structure is visible. Finally use Tool #3 (Eliminate Possibilities) to confirm the winning value against the answer choices and to check that the tempting -20 is actually out of reach.

1STEP 1

Factor the expression

Both terms of 2a-ab share the factor a, so the whole expression is the single product a(2-b).

2a-ab=a(2-b)
2STEP 2

Decide the sign

Since a is at least 1, the product turns negative only when b is greater than 2, and then a(2-b) equals -a(b-2).

b ＞ 2 → a(2-b)=-a(b-2)
3STEP 3

Push both factors to the edge

Take the largest allowed values a=5 and b=5, and the expression gives 2·5-5·5=10-25=-15.

a=b=5: 2·5-5·5=10-25=-15
4STEP 4

Confirm against the choices

a(b-2) tops out at 5·3=15, so nothing sinks below -15; the tempting -20 would need a(b-2)=20, leaving choice (B).

a(b-2) ≤ 5·3=15 → 2a-ab ≥ -15 → (B)
Answer
-15
The result -15 comes from the corner a=b=5, which matches the intuition that a minimum lives at a boundary. It clears the built-in trap: -20 looks reachable only if you forget that a and b both cap at 5, so a(b-2) cannot exceed 5·3=15. A quick scan of nearby corners agrees — a=4,b=5 gives -12 and a=5,b=4 gives -10, both larger than -15 — so -15 really is the lowest.
💡Key takeaway

For a smallest-value question over a small range, factor the expression and push each letter to the edge of what it's allowed to be.

  • Factor the expression
  • Decide the sign
  • Push both factors to the edge
  • Confirm against the choices

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