AMC 10 · 2020 · #4

Grade 8 geometry-2d

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

angle-sum-triangleprime-numbersprimality-test guess-and-checksystematic-enumeration ↑ Prerequisites: prime-numbersangle-sum-triangle

📏 Short solution 💡 2 insights

Problem

A right triangle has acute angles of a° and b° with a ＞ b, both prime. Find the smallest value b can take.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

Tool #3 (Eliminate Possibilities): only five candidate values for b are offered. Test each in order from smallest, check whether 90 - b is also prime — first success wins. Tool #6 (Guess and Check): we test smallest first because we want the minimum. Tool #16 (Change Focus): switching the question "is b prime?" into the easier question "is 90 - b prime?" lets each candidate be settled with one quick primality check.

1STEP 1

A triangle's angles sum to 180°; drop the 90° right angle and the two acute angles give a + b = 90.

a + b = 90

2STEP 2

Try b = 2, but a = 88 = 8 × 11 is composite — eliminate.

b = 2 → a = 88 (composite) ✗

3STEP 3

Try b = 3, but a = 87 = 3 × 29 is composite — eliminate.

b = 3 → a = 87 = 3 · 29 ✗

4STEP 4

Try b = 5, but a = 85 = 5 × 17 is composite — eliminate.

b = 5 → a = 85 = 5 · 17 ✗

5STEP 5

Try b = 7: a = 83 is prime and 83 ＞ 7, so this pair works — the smallest that does.

b = 7 → a = 83 prime ✓ → (D)

Answer

83 + 7 = 90 confirms the angle sum, both are prime, and 83 ＞ 7. We rejected b = 2, 3, 5 because each produced a composite a, so 7 is indeed the smallest.

💡Key takeaway

This AMC 10 problem only needs Grade 8 “triangle angles add to 180°” plus a quick prime check you already know — try b = 2, 3, 5, 7 in order and b = 7 is the first that pairs with a prime (83).

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