2020 AMC 10B Problem 4

1. 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 $90^\circ$: $a + b = 90$.
2. 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.

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

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

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.

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