AMC 10 · 2020 · #4
Grade 8 geometry-2dPick an answer.
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.
Write the angle sum
The two acute angles add to ninety.
Subtracting the right angle from the total leaves 90° for the other two angles to share.
Subtracting the right angle from the total leaves ninety degrees for the other two angles to share.
▸ Why?
The three angles of a triangle always add to a straight angle, whatever the shape.
▸ Why?
A right angle is exactly half a straight angle, so what remains is the other half.
Test the smallest prime
Its partner is composite, so it fails.
Even numbers greater than 2 are not prime — fastest rule to apply.
4.OA.B.4Guess And CheckTest the next prime
Again the partner is composite.
Digits of 87 sum to 15, a multiple of 3, so 87 is divisible by 3.
4.OA.B.4Guess And CheckTest the one after
It still fails.
85 ends in 5, so divisible by 5.
4.OA.B.4Guess And CheckFind the one that works
The first with a prime partner is 7.
Since smaller choices all failed, b = 7 is the smallest prime that works.
4.OA.B.4Eliminate PossibilitiesThis AMC 12 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).