AMC 10 · 2002 · #11
Grade 4 number-theoryPick an answer.
Parity (odd/even) and the single even prime pin the numbers down so tightly that only one set of values survives. So the smart move is to rule out cases with even/odd reasoning and divisibility, until just one answer for A and B remains, then read off the sum.
Name the four primes
Write the four plainly; since the difference must be a positive prime, A exceeds B.
Getting the four objects and their relationships on paper is what makes the hidden even/odd clue visible.
2.OA.C.3Introduce A VariableBoth odd is impossible
If both were odd the sum and difference would both be even, impossible, so one is 2.
Odd minus odd is even, and 2 is the lone even prime, so two big even primes simply cannot both exist here.
The two primes cannot both be odd, because an odd difference forces one of them to be even.
▸ Why?
Odd minus odd is even, so an odd gap can only come from one odd and one even number.
▸ Why?
Two is the only even prime, so the even one of the pair has no choice about what it is.
Decide which one is 2
A cannot be 2 or the difference would not be positive, so B = 2.
Only the smaller number can be 2, because subtracting from 2 would drop below the primes.
4.OA.B.4Extreme PrincipleUse divisibility by 3
Then A-2, A, A+2 cover all remainders mod 3, forcing the smallest to be 3 and A = 5.
Three numbers two apart always hit every remainder mod 3, so one is forced to be the prime 3.
4.OA.B.4Eliminate PossibilitiesAdd them and pick the property
The four primes total 17, which is itself prime, choice (E).
Once the only surviving primes are 2, 3, 5, 7, their total 17 is itself prime.
4.OA.B.4Eliminate PossibilitiesSince 2 is the only even prime, B has to be 2, and then divisibility by 3 forces A = 5, giving the primes 2, 3, 5, 7 whose sum 17 is itself prime.
- Name the four primes
- Both odd is impossible
- Decide which one is 2
- Use divisibility by 3
- Add them and pick the property