AMC 8 2019 Problem 23

AMC8 2019 年第 23 题

整除与因数★★★☆☆

After Euclid High School's last basketball game, it was determined that

\displaystyle \frac{1}{4}

of the team's points were scored by Alexa and

\displaystyle \frac{2}{7}

were scored by Brittany. Chelsea scored 15 points. None of the other 7 team members scored more than 2 points. What was the total number of points scored by the other 7 team members?

在 Euclid 高中最近的篮球赛后，发现队伍总分的

\displaystyle \frac{1}{4}

是 Alexa 获得的，

\displaystyle \frac{2}{7}

是 Brittany 获得的，Chelsea 得了 15 分。队伍中的其他 7 个队员没有一个得分超过 2 分。问其他 7 个队员获得的总分是多少分？

A.10
B.11
C.12
D.13
E.14

正确答案 · Correct Answer

解析 · Solution

Solution 1

Given the information above, we start with the equation

\displaystyle \frac{t}{4} + \frac{2t}{7} + 15 + x = t

, where

t

is the total number of points scored and

x\le 14

is the number of points scored by the remaining 7 team members, we can simplify to obtain the Diophantine equation

\displaystyle x+15 = \frac{13}{28} t

, or

28x+28\cdot 15=13t

. Since

t

is necessarily divisible by 28, let t=28u where

u \ge 0

and divide by 28 to obtain x + 15 = 13u. Then, it is easy to see u=2 ( t=56 ) is the only candidate remaining, giving x=11. -scrabbler94 Edits by Wrenmath

We first start by setting the total number of points as 28, since

\text{LCM}(4,7) = 28

. However, we see that this does not work since we surpass the number of points just with the information given (

\displaystyle 28\cdot \frac{1}{4} +28\cdot \frac{2}{7} + 15 = 30

(> 28) ). Next, we can see that the total number of points scored is 56 as, if it is more than or equal to 84, at least one of the others will score more than 2 points. With this, we have that Alexa, Brittany, and Chelsea score:

\displaystyle 56\cdot \frac{1}{4} +56\cdot \frac{2}{7} + 15 = 45

, and thus, the other seven players would have scored a total of 56-45 = 11. (We see that this works since we could have 4 of them score 2 points, and the other 3 of them score 1 point.) -aops5234 -Edited by Penguin_Spellcaster — Modular Arithmetic Adding together Alexa's and Brittany's fractions, we get

\displaystyle \frac{15}{28}

as the fraction of the total number of points they scored together. However, this is just a ratio, so we can introduce a variable:

\displaystyle \frac{15x}{28x}

where

x

is the common ratio. Let

y

and

z

and

w

be the number of people who scored 1, 2, and 0 points, respectively. Writing an equation, we have

\displaystyle \frac{13x}{28x} = 15 + y + 2z + 0w.

We want all of our variables to be integers. Thus, we want

15 + y + 2z = 0 \pmod {13}.

Simplifying,

y+2z = 11 \pmod {13}.

The only possible value, as this integer sum has to be less than

7 \cdot 2 + 1 = 15,

must be 11. Therefore, y+2z = 11, and the answer is

\textbf{(B) 11}

. - ab2024: Answer choices We can rewrite the question as an algebraic equation:

\displaystyle \frac{1}{4} T + \frac{2}{7} T + 15 + y = x

, where

x

represents the total amount of points and

y

the amount of points the 7 other players scored. From there, we add the two fractions to get

\displaystyle \frac{15}{28} x + 15 + y = x

. Subtracting

\displaystyle \frac{15}{28} x

from both sides, we get

\displaystyle \frac{13}{28} x = y + 15

. We multiply each side by 28 to get rid of the denominator, in which we get 13x = 420 + 28y. Now let's think of this logically. This equation is telling us that if you add 420 and 28 times the amount of points scored by the extra 7 players, you get 13 times the amount of points total. And since we have to have a whole number of points total, this means that 420 + 28y must be divisible by 13. Plugging in all the answer choices for

y

, we find that the only answer that makes 420 + 28y divisible by 13 is

\textbf{(B) 11}

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