2020 AMC 10B Problem 4

P1-10~2 minPrint

Primes Angles & Polygons Organized listing / direct enumeration

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 $b$ ?

Show solution without answering

Key insight

The two acute angles add to 90. Try b = 2, 3, 5, 7 in order and stop at the first one where 90 minus b is also prime.

Solution

In a right triangle the three angles add to $180^\circ$ and one of them is $90^\circ$ . So the two acute angles add to $90^\circ$ : $a + b = 90$ .
We want the smallest prime $b$ for which $a = 90 - b$ is also prime. Since we want the smallest, start at the smallest prime and go up.

$b$	$a = 90 - b$	Is $a$ prime?
$2$	$88$	No, it is even
$3$	$87$	No, $87 = 3 \times 29$
$5$	$85$	No, $85 = 5 \times 17$
$7$	$83$	Yes

At $b = 7$ we get $a = 83$ , which is prime, and $83 > 7$ as required. No smaller $b$ worked.

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

Why this works

For a "least possible" question with only a few candidates, test them from smallest upward and stop at the first one that works. A quick shortcut: $90$ is even, so if $b = 2$ then $a$ is even too and cannot be prime. That is why the smallest prime fails right away.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(D)}\ 7}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 2: Primes and Composites
Introduction to Geometry, Chapter 2: Angles
Prealgebra, Chapter 15: Problem-Solving Strategies (§15.2)
Introduction to Number Theory, Chapter 4: Prime Factorization (§4.2, §4.4)