When each of the athletes at a sports event was given the same number

Question

When each of the athletes at a sports event was given the same number of sports drinks from a crate containing 80 sports drinks, the number of sports drinks left in the crate was 4 less than the number of athletes at the sports event. Which of the following CANNOT be the number of athletes at the sports event?

A. 12
B. 14
C. 21
D. 24
E. 28

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Attachment:

2023-12-01_21-10-02.png

Bunuel · Accepted Answer

Let's denote the number of athletes as x. The problem states that when each athlete receives an  equal number  of drinks from a crate containing 80 drinks, the number of remaining drinks is 4 less than the number of athletes. This essentially sets up a remainders problem: 80 divided by a divisor x gives a remainder of x - 4, which can be written as:

80 = qx + (x - 4), where x - 4 ≥ 0;

84 = x(q + 1);

q + 1 = 84/x.

Since q + 1 must be an integer, 84/x must also be an integer, meaning that x, the number of athletes, must be a factor of 84. The only option that is not a factor of 84 is D, 24.

Answer: D.

gmatophobia · Answer

We can backward, i.e. use the options to our advantage to find the solution Screenshot 2023-11-17 172236.jpg From the number of drinks consumed, we can find the number of drinks left. The number of drinks left should be divisible by the number of members in the team. Option D

Vaishvii · Answer

left (80 - athletes*bottles given to each player) = athletes - 4

80-ax = a-4 
substitute a with different values, all numbers except for 24 return int x

rakman123 · Answer

The 80 bottles need to be  equally  divided between the athletes, with a remainder of 4. So just  add the remainder to the total number of bottles  and you get  84 .

lets have a= athletes & i= integer

84/a = i

Using the options in the answer bank, plug them in for 'a' athletes, and see what gives you an 'i' integer.

Voila.

sachi-in · Answer

If let there be n players and each receive a number of items then -

n * a + ( n - 4 ) = 80 ( n - 4 items left )

n*( a + 1) = 84 = 2^2 * 3 * 7

We can break the 84 into two factors n and ( a + 1 ) 
but ofcourse ( a + 1 ) > 1 since each person should have received atleast 1 items each , or a >= 1

2*42 , 3*28 , 4*21 , 6*14 , 12*7 ... all but 24 (3*8) since highest power of 2 is 2 here

MBAToronto2024 · Answer

KarishmaB 
I was using backtracking. And it took too long. I thought about short-cut: 
80/21 gives remainder of 17/21. Logically, 80/24 will give remainder of 20/24. But that's not the case. It's a trap.

KarishmaB · Answer

80/21 gives a remainder of 17. - Correct
Logically, 83/21 will give a remainder of 20 (not 80/24). I am adding 3 more balls to the total number of balls (to the dividend), so remainder will increase by 3. I cannot add 3 more balls to each group i.e. to the divisor. 
Hence that logic did not work.

These concepts of division are discussed here:  https://youtu.be/A5abKfUBFSc

KarishmaB · Answer

Backsolving is the best here:

If n = 12, 8 drinks will be leftover. Of 80 drinks, 72 were equally distributed among 12 people and 8 were left. Works.
If n = 14, 10 drinks will be leftover. Of 80 drinks, 70 were equally distributed among 14 people and 10 were left. Works.
n = 21 - looks like too much work. Let's go to 24 and 28 first. Far easier numbers.
If n = 24, 20 drinks will be leftover. Of 80 drinks, 60 were equally distributed among 24 people. Not possible.

Answer (D)

HarshavardhanR · Answer

Say,
Number of athletes = a
Number of sports drinks per athlete = n

What we are given, when translated ->

(an) + remainder = 80
(an) + (a-4) = 80
=> a(1+n) = 84

Number of athletes (a) has to be a factor of 84. Out of the choices, only choice D (24) is not a factor. It is our answer.

Harsha

shubhim20 · Answer

why q+1 has to be an integer?

Bunuel · Answer

Because q is the quotient when dividing 80 by x, it must be an integer. So q + 1 must also be an integer. That means 84/x must be an integer too.

GMATinsight · Answer

Let, Atheletes = a
Remaining Drinks = a-4

i.e. Distributed drinks = 80 - (a-4) = 84 -a

84 - a is Divsible by a

i.e.  \frac{84 - a }{ a} = Integer 
i.e.  \frac{84 }{ a} - \frac{a}{a} = Integer

i.e.  \frac{84 }{ a} -  1  = Integer

i.e. 84 is divisible by a
i.e. a = factor of 84

factors of 84 are {1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84}

out of give options only 24 is not a factor of 84

Option D

Related Video:  https://www.youtube.com/watch?v=FWm0f4YnNGc

When each of the athletes at a sports event was given the same number

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