Competition · AMC preparation · step 4 of 4
AMC 8 · 2014 · #4
Grade 4 number-theoryPick an answer.
AMC 8 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The sum 85 is odd, and odd + odd = even while even + even = even. So one of the two primes must be even and the other odd — that is a powerful elimination (Tool #3). The only even prime is 2, so one prime is forced to be 2 and the other is 85 - 2 = 83. Tool #2 (Systematic List) is the backup: we can also list small primes 2, 3, 5, 7, 11, …, pair each with 85 - p, and check primality — both routes land on the same pair.
Use parity to narrow it down
The sum 85 is odd, but odd+odd and even+even are both even — so one prime is even and the other odd.
Knowing odd + even = odd is a Grade 2 fact, and we use it to rule out every pair of two odd primes in one shot.
2.OA.C.3Eliminate PossibilitiesFind the only even prime
The only even prime is 2 — every even number past 2 is divisible by 2 — so one prime must be 2.
Identifying 2 as the lone even prime is a Grade 4 prime/composite fact and finishes the elimination.
One of the two primes is forced to be the number 2.
▸ Why?
The two primes cannot both be odd: if they were, their sum would be even, but 85 is odd, so one prime must be even and the other odd.
▸ Why?
Two odd numbers always add up to an even number, so a pair of odd primes could never reach the odd total 85.
▸ Why?
Each odd number is a whole count of twos with a single one left over, so when two odds are added the two leftover ones join into one more group of two, leaving the total made entirely of twos, which is even.
▸ Why?
The even prime among them can only be 2, because every even number larger than 2 fails to be prime.
▸ Why?
Any even number bigger than 2 splits into a whole number of groups of two, so 2 divides it evenly; that gives it a divisor other than 1 and itself, which a prime is not allowed to have.
Subtract to get the other prime
Subtract to get the other prime: 85 - 2 = 83.
A single subtraction pins down the second number; the only remaining job is to check it really is prime.
4.OA.B.4Eliminate PossibilitiesCheck 83 for primality
Check 83 is prime: √83 ＜ 10, so test 2, 3, 5, 7 — none divide it, so 83 is prime.
Listing the primes up to √(83) in order is a tiny Tool #2 list — only four numbers to test.
4.OA.B.4Make A Systematic ListMultiply the two primes
Multiply the two primes: 2 × 83 = 166, choice (E).
2 × 83 is a Grade 3 multiplication; doubling 83 gives 166 and matches choice (E).
3.OA.C.7Eliminate PossibilitiesWhenever two primes add to an odd number, one of them has to be 2 — that single Grade 4 parity trick cracks the whole problem.
- Use parity to narrow it down
- Find the only even prime
- Subtract to get the other prime
- Check 83 for primality
- Multiply the two primes
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