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2020 AMC 10B Problem 4

2020 AMC 10B Problem 4AMC 10 Step by Step

Name: Date: September 13, 2026

Primes · difficulty 1 of 5 · about 2 min · position P1-10

The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$ , where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$ ?

A)
$2$
B)
$3$
C)
$5$
D)
$7$
E)
$11$

Answer

Key insight

The angles sum to 90, so test the primes b = 2, 3, 5, 7 in order until 90 - b is also prime.

Solution

In a right triangle the two acute angles satisfy $a + b = 90$ . We want the smallest prime $b$ for which $a = 90 - b$ is also prime (and automatically $a > b$ ).

Check primes from the bottom:

$b = 2$ : $a = 88$ , even, not prime.
$b = 3$ : $a = 87 = 3 \cdot 29$ , not prime.
$b = 5$ : $a = 85 = 5 \cdot 17$ , not prime.
$b = 7$ : $a = 83$ , which is prime.

So the least possible value is $b = 7$ .

The answer is $\boxed{\textbf{(D)}\ 7}$ .

Why this works

"Least possible" with a small search space means: start at the smallest candidate and test upward, stopping at the first success. Note the parity shortcut: since $90$ is even, $b = 2$ would force $a = 88$ , so both primes must be odd, which is why $2$ fails immediately.

The trap

Stopping at b = 2 because 2 is the smallest prime, without checking that 88 is not prime.

Common mistakes

Stopping at b = 2 because 2 is the smallest prime, without checking that 88 is not prime.
Misjudging $87$ or $85$ as prime; both have small factors ( $87 = 3 \cdot 29$ , $85 = 5 \cdot 17$ ).

Techniques

Organized listing / direct enumeration