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

In Theresa's first $8$ basketball games, she scored $7, 4, 3, 6, 8, 3, 1$ and $5$ points. In her ninth game, she scored fewer than $10$ points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than $10$ points and her points-per-game average for the $10$ games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: First $8$ scores: $7, 4, 3, 6, 8, 3, 1, 5$; Game $9$ score is a whole number from $0$ to $9$; Game $10$ score is a whole number from $0$ to $9$; The average over $9$ games is an integer; The average over $10$ games is an integer; Answer choices: (A) $35$, (B) $40$, (C) $48$, (D) $56$, (E) $72$

Unknowns: The product (game $9$ score) $\times$ (game $10$ score)

Understand

Plan

Primary tool: #12 Use Modular Arithmetic

Secondary: #7 Break into Subproblems

"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.

Execute — Answer: B

#7 Break into Subproblems 4.OA.A.3 Step 1

Add up the first $8$ scores to get the starting total.

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

💡 Before any divisibility thinking, just compute the running total — Grade 4 multi-step arithmetic.

#12 Use Modular Arithmetic 6.SP.B.5 Step 2

Translate "the $9$-game average is an integer" into a divisibility statement.
The average is total $\div 9$, so the $9$-game total must be a multiple of $9$.

$$37 + (\text{game }9) \equiv 0 \pmod{9}$$

💡 Mean $=$ sum $\div$ count, so an integer mean forces the sum to be divisible by the count.

#12 Use Modular Arithmetic 4.OA.B.4 Step 3

Find the smallest multiple of $9$ that is at least $37$.
The multiples of $9$ near $37$ are $36$ and $45$.
Since the game-$9$ score is non-negative, the new total must be $\geq 37$, and the cap (game $9$ $< 10$) means the total can rise by at most $9$, so it must be $45$.

$$\text{multiples of }9: \ldots, 36, 45, 54, \ldots \;\Rightarrow\; \text{new total} = 45$$

💡 Grade 4 "find multiples in a range": only $45$ lies in $[37,\,37+9]$.

#7 Break into Subproblems 4.OA.A.3 Step 4

Subtract to read off the game-$9$ score.

$$\text{game }9 = 45 - 37 = 8 \;\;(\text{and } 8 < 10 \checkmark)$$

💡 First subproblem closed: she scored $8$ in game $9$.

#12 Use Modular Arithmetic 4.OA.B.4 Step 5

Now repeat the same idea for game $10$.
The $10$-game total must be a multiple of $10$, and the new total is $45 + (\text{game }10)$ with game $10$ between $0$ and $9$.
The only multiple of $10$ in the interval $[45,\,54]$ is $50$.

$$45 + (\text{game }10) \equiv 0 \pmod{10} \;\Rightarrow\; \text{new total} = 50$$

💡 Multiples of $10$ end in $0$, so once the total passes $45$ the next stop is $50$.

#7 Break into Subproblems 4.OA.A.3 Step 6

Subtract again to read off the game-$10$ score, then multiply the two game scores to answer the question.

$$\text{game }10 = 50 - 45 = 5,\quad 8 \times 5 = 40 \;\Rightarrow\; \textbf{(B)}$$

💡 Second subproblem closed, and the final product is the answer.

[1] #7 4.OA.A.3 Add up the first $8$ scores to get the starting total.

[2] #12 6.SP.B.5 Translate "the $9$-game average is an integer" into a divisibility statement. Th

[3] #12 4.OA.B.4 Find the smallest multiple of $9$ that is at least $37$. The multiples of $9$ ne

[4] #7 4.OA.A.3 Subtract to read off the game-$9$ score.

[5] #12 4.OA.B.4 Now repeat the same idea for game $10$. The $10$-game total must be a multiple o

[6] #7 4.OA.A.3 Subtract again to read off the game-$10$ score, then multiply the two game score

Review

Reasonableness: Check both averages. After $9$ games the total is $45$, so the average is $45 \div 9 = 5$, an integer. After $10$ games the total is $45 + 5 = 50$, so the average is $50 \div 10 = 5$, also an integer. Both game-$9$ and game-$10$ scores ($8$ and $5$) are below $10$, as required. The product $8 \times 5 = 40$ 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.

Alternative: Tool #6 (Guess and Check): try each game-$9$ score $0,1,2,\ldots,9$. The totals are $37,38,\ldots,46$, and only $45$ is divisible by $9$, giving game $9 = 8$. Then try each game-$10$ score $0,1,\ldots,9$. The totals are $45,46,\ldots,54$, and only $50$ is divisible by $10$, giving game $10 = 5$. Product $= 40$. Slower than the divisibility argument but the same answer.

CCSS standards used (min grade 6)

4.OA.A.3 Solve multistep word problems with whole numbers using the four operations (Adding the first $8$ scores to get $37$, and subtracting to recover each game's score from the running total.)
4.OA.B.4 Find all factor pairs; recognize multiples (Locating the next multiple of $9$ above $37$ (namely $45$) and the next multiple of $10$ above $45$ (namely $50$).)
6.SP.B.5 Summarize numerical data sets, including reporting the number of observations and measures of center (Translating "average over $n$ games is an integer" into the divisibility condition "running total is a multiple of $n$".)

⭐ 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.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

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