AMC 10 · 2020 · #4
Grade 8 geometry-2dProblem
The acute angles of a right triangle are a∘ and b∘, where a>b and both a and b are prime numbers. What is the least possible value of b?
Pick an answer.
AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A right triangle has acute angles of $a^\circ$ and $b^\circ$ with $a > b$, both prime. Find the smallest value $b$ can take.
Givens: The triangle is a right triangle (one angle is $90^\circ$); The two acute angles measure $a^\circ$ and $b^\circ$; $a + b = 90$ (acute angles of a right triangle sum to $90^\circ$); Both $a$ and $b$ are prime numbers, $a > b$; Answer choices: (A) $2$, (B) $3$, (C) $5$, (D) $7$, (E) $11$
Unknowns: The minimum possible value of $b$
Understand
Restated: A right triangle has acute angles of $a^\circ$ and $b^\circ$ with $a > b$, both prime. Find the smallest value $b$ can take.
Givens: The triangle is a right triangle (one angle is $90^\circ$); The two acute angles measure $a^\circ$ and $b^\circ$; $a + b = 90$ (acute angles of a right triangle sum to $90^\circ$); Both $a$ and $b$ are prime numbers, $a > b$; Answer choices: (A) $2$, (B) $3$, (C) $5$, (D) $7$, (E) $11$
Plan
Primary tool: #3 Eliminate Possibilities
Secondary: #6 Guess and Check, #16 Change Focus / Count the Complement
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.
Execute — Answer: D
8.G.A.5 Step 1 - Use the angle-sum fact.
- The three angles of a triangle sum to $180^\circ$.
- With one $90^\circ$ angle, the two acute angles satisfy $a + b = 180 - 90 = 90$.
💡 Subtracting the right angle from the total leaves $90^\circ$ for the other two angles to share.
4.OA.B.4 Step 2 - Try the smallest choice, $b = 2$.
- Then $a = 90 - 2 = 88$.
- But $88 = 8 \times 11$ is even and composite, so $a$ is not prime.
- Eliminate $b = 2$.
💡 Even numbers greater than $2$ are not prime — fastest rule to apply.
4.OA.B.4 Step 3 - Try $b = 3$.
- Then $a = 90 - 3 = 87 = 3 \times 29$, composite.
- Eliminate.
💡 Digits of $87$ sum to $15$, a multiple of $3$, so $87$ is divisible by $3$.
4.OA.B.4 Step 4 - Try $b = 5$.
- Then $a = 90 - 5 = 85 = 5 \times 17$, composite.
- Eliminate.
💡 $85$ ends in $5$, so divisible by $5$.
4.OA.B.4 Step 5 - Try $b = 7$.
- Then $a = 90 - 7 = 83$.
- Check primality of $83$: test divisibility by primes up to $\sqrt{83} \approx 9.1$, namely $2, 3, 5, 7$.
- None divide $83$, so $83$ is prime.
- Also $83 > 7$, so the condition $a > b$ holds.
💡 Since smaller choices all failed, $b = 7$ is the smallest prime that works.
8.G.A.5 Use the angle-sum fact. The three angles of a triangle sum to $180^\circ$. With 4.OA.B.4 Try the smallest choice, $b = 2$. Then $a = 90 - 2 = 88$. But $88 = 8 \times 11$ 4.OA.B.4 Try $b = 3$. Then $a = 90 - 3 = 87 = 3 \times 29$, composite. Eliminate. 4.OA.B.4 Try $b = 5$. Then $a = 90 - 5 = 85 = 5 \times 17$, composite. Eliminate. 4.OA.B.4 Try $b = 7$. Then $a = 90 - 7 = 83$. Check primality of $83$: test divisibility Review
Reasonableness: $83 + 7 = 90$ 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.
Alternative: Tool #2 (Systematic List) — list primes under $45$: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43$. Pair each with $90 - b$ and check both are prime. The first hit is $(7, 83)$.
CCSS standards used (min grade 8)
4.OA.B.4Find all factor pairs and recognize multiples; determine prime or composite (Testing each candidate $a = 90 - b$ for primality.)8.G.A.5Use informal arguments to establish facts about angle sum and exterior angles (Justifying $a + b = 90$ from the triangle's angle sum and the right angle.)
⭐ This AMC 10 problem only needs Grade 8 “triangle angles add to $180^\circ$” 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 ($83$).
⭐ This AMC 10 problem only needs Grade 8 “triangle angles add to $180^\circ$” 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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