Competition · AMC preparation · step 4 of 4

AMC 8 · 2008 · #15

Grade 6 arithmeticnumber-theory
Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration
divisibility-rulesmodular-arithmeticmean-median-mode-rangesystematic-enumeration modular-arithmeticsystematic-enumeration ↑ Prerequisites: divisibility-rulesmean-median-mode-range
📏 Medium solution 💡 3 insights
Problem
Theresa scored 7, 4, 3, 6, 8, 3, 1, 5 in her first 8 games. In game 9 she scored fewer than 10 points and her 9-game average came out to an integer. In game 10 she also scored fewer than 10 points and her 10-game average was an integer too. Find the product of her game-9 and game-10 scores.

Pick an answer.

(A)
35
(B)
40
(C)
48
(D)
56
(E)
72

AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Use Modular Arithmetic

"Average is an integer" is just a divisibility statement in disguise: the running total has to be a multiple of the number of games. That makes Tool #12 (divisibility/modular arithmetic) the right lens. Tool #7 splits the work into two clean subproblems — first pin down the game-9 score using divisibility by 9, then use that result to pin down the game-10 score using divisibility by 10. The "fewer than 10" cap is what forces a unique answer in each step.

1STEP 1

Add the first 8 scores

Add up the first 8 scores: the running total is 37.

7+4+3+6+8+3+1+5 = 37
2STEP 2

Turn the average into divisibility

"9-game average is an integer" just means the 9-game total must be a multiple of 9.

37 + (game 9) ≡ 0 (mod 9)
3STEP 3

Find the next multiple of 9

The only multiple of 9 in the reachable range [37, 46] is 45, so the new total is 45.

multiples of 9: …, 36, 45, 54, … → new total = 45
4STEP 4

Subtract for game 9

Subtract to read off the game-9 score: 8 (and 8 is below 10).

game 9 = 45 - 37 = 8 (and 8 ＜ 10 ✓)
5STEP 5

Repeat for game 10

Repeat for game 10: the only multiple of 10 in [45, 54] is 50, so the total becomes 50.

45 + (game 10) ≡ 0 (mod 10) → new total = 50
6STEP 6

Multiply the two scores

The game-10 score is 5; multiply the two scores: 8 × 5 = 40.

game 10 = 50 - 45 = 5, 8 × 5 = 40 → (B)
Answer
40
Check both averages. After 9 games the total is 45, so the average is 45 ÷ 9 = 5, an integer. After 10 games the total is 45 + 5 = 50, so the average is 50 ÷ 10 = 5, also an integer. Both game-9 and game-10 scores (8 and 5) are below 10, as required. The product 8 × 5 = 40 matches choice (B). Sanity check the uniqueness: any larger multiple of 9 above 37 would be 54, requiring a game-9 score of 17, which breaks the " ＜ 10" cap — so 45 really is forced.
💡Key takeaway

An integer average is just a multiple in hiding. Hop to the next multiple of 9, then to the next multiple of 10 — the two hops give the missing scores.

  • Add the first 8 scores
  • Turn the average into divisibility
  • Find the next multiple of 9
  • Subtract for game 9
  • Repeat for game 10
  • Multiply the two scores

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