Competition · AMC preparation · step 4 of 4
AMC 10 · 2024A · #3
Grade 4 number-theoryPick an answer.
AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
To make P as small as possible, we want the five primes themselves to be as small as possible — that points straight at Tool #6 (Guess and Check). We try the smallest candidate set first, check whether the sum is prime, and if not, swap in the next prime up. Before guessing, Tool #3 (Eliminate Possibilities) rules out using 2: if 2 is one of the five, the other four are odd, the sum is even, and the only even prime is 2 itself — way too small. So all five must be odd primes, which narrows the search to a tiny list.
Rule out the prime 2
Rule out the prime 2: using it makes the sum even and at least 28, so never prime — all five primes must be odd.
Adding four odd numbers gives an even number, and adding 2 keeps it even. Even numbers above 2 are never prime — Grade 2 odd/even rules eliminate the case.
Adding four odd numbers gives an even total, and adding two keeps it even.
▸ Why?
Odd numbers pair off two at a time into even sums, so an even count of them adds to an even total.
▸ Why?
An even number above two always carries the factor two, so it can never be prime.
Try the five smallest odd primes
First guess the five smallest odd primes 3, 5, 7, 11, 13 — the smallest possible sum is 39.
Starting from the smallest possible numbers guarantees the smallest possible sum — there is nowhere lower to look.
4.NBT.B.4Guess And CheckTest 39 for primality
Is 39 prime? Its digits 3+9=12 is a multiple of 3, so 39 = 3 × 13 is composite — try the next sum.
Divisibility by 3 is fast: digits 3 + 9 = 12 is a multiple of 3, so 39 is too.
4.OA.B.4Guess And CheckSwap in the next prime
Cheapest bump: swap the largest prime 13 for 17, raising the sum by 4 to 43.
Trading 13 for 17 adds 4 to the sum: 39 + 4 = 43. That is the smallest possible next sum.
4.NBT.B.4Guess And CheckTest 43 for primality
Check 43: not divisible by 2, 3, or 5 (all primes up to √43 ≈ 6.6), so 43 is prime.
Only need to check primes up to √(43) ≈ 6.6, so 2, 3, 5 suffices.
4.OA.B.4Guess And CheckRule out 41
Only 41 lies between 39 and 43, but 41 - (3+5+7+11) = 15 isn't prime, so 43 is smallest.
Only 41 sits between 39 and 43, and it cannot be hit by any allowed combination — so 43 is locked in.
4.OA.B.4Eliminate PossibilitiesAdd the digits of 43
Add the digits of 43: 4 + 3 = 7 → (B).
Reading the tens digit (4) and the ones digit (3) and adding them is a Grade 2 place-value move.
2.NBT.A.1Guess And CheckThrow out the prime 2 on parity grounds, add the five smallest odd primes, and bump the largest one up until the sum is prime. The answer 43 falls out on the second try, and its digits add to 7.
- Rule out the prime 2
- Try the five smallest odd primes
- Test 39 for primality
- Swap in the next prime
- Test 43 for primality
- Rule out 41
- Add the digits of 43
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