Competition · AMC preparation · step 4 of 4

AMC 8 · 2014 · #23

Grade 6 number-theorylogic
Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration
prime-numberslogical-deductionbound-inequality-then-enumerate bound-inequality-then-enumeratesystematic-enumerationcasework ↑ Prerequisites: prime-numberslogical-deduction
📏 Medium solution 💡 4 insights
Problem
Ashley, Bethany, and Caitlin each wear a 2-digit prime number on their softball uniforms. The sum of any two of their numbers is a calendar date in the current month: Ashley + Caitlin = Bethany's birthday (earlier this month), Ashley + Bethany = Caitlin's birthday (later this month), and Bethany + Caitlin = today's date (between the two birthdays). Find Caitlin's uniform number.

Pick an answer.

(A)
11
(B)
13
(C)
17
(D)
19
(E)
23

AMC 8 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Eliminate Possibilities

The candidate set is tiny: 2-digit primes whose pairwise sums fit on a calendar (at most 31). Tool #2 (Systematic List) lets us write that short list explicitly, and Tool #3 (Eliminate) knocks out every prime ≥ 19 because pairing it with any other 2-digit prime gives a sum past 31. Tool #7 (Identify Subproblems) splits the question into two cleaner pieces: (a) which three primes are used, and (b) which girl wears which number. Part (b) is decided by chaining the date inequalities, never algebra.

1STEP 1

List the two-digit primes

List the 2-digit primes: 11, 13, 17, 19, 23, 29, … — we only need the smallest few, since any pairwise sum must stay ≤ 31.

2-digit primes: 11, 13, 17, 19, 23, 29, 31, …
2STEP 2

Drop the primes that are too big

Any prime ≥ 19 fails: 13 + 19 = 32 ＞ 31. Only 11, 13, 17 survive, and three distinct primes are needed, so the set must be {11, 13, 17}.

13 + 19 = 32 ＞ 31 → 19 (and larger) cannot appear → set = {11, 13, 17}
3STEP 3

Check the three pairwise sums

Check {11, 13, 17} works: 11 + 13 = 24, 11 + 17 = 28, 13 + 17 = 30 — all ≤ 31 and distinct, so they match the three different dates.

Pairwise sums: 11+13 = 24, 11+17 = 28, 13+17 = 30
4STEP 4

Order the three numbers

Let A, B, C be their numbers. The date order A + C ＜ B + C ＜ A + B; cancel the shared term in each: A ＜ B and C ＜ A, so C ＜ A ＜ B.

A + C ＜ B + C ＜ A + B ⟹ C ＜ A ＜ B
5STEP 5

Assign a prime to each girl

By C ＜ A ＜ B, Caitlin gets the smallest, Ashley the middle, Bethany the largest: C = 11, A = 13, B = 17. So Caitlin wears 11 → (A).

C = 11, A = 13, B = 17 → Caitlin = 11 → (A)
Answer
11
Plug back in: Bethany's birthday = A + C = 13 + 11 = 24, today = B + C = 17 + 11 = 28, Caitlin's birthday = A + B = 13 + 17 = 30. The dates 24 ＜ 28 ＜ 30 all fall in the same month, and the story's order (Bethany earlier, today now, Caitlin later) is satisfied. Every sum is ≤ 31, so every date is real. The answer 11 is the smallest answer choice, which matches the deduction that Caitlin wears the smallest of the three primes.
💡Key takeaway

This AMC 8 problem only needs the Grade 6 idea that you can subtract the same number from both sides of an inequality — the prime-number list and date cap do all the rest of the work!

  • List the two-digit primes
  • Drop the primes that are too big
  • Check the three pairwise sums
  • Order the three numbers
  • Assign a prime to each girl

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