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)

(B)

(C)

(D)

(E)

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

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

2STEP 2

"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

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 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: 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

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

game 10 = 50 - 45 = 5, 8 × 5 = 40 → (B)

Answer

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.

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