How many crayons are in a certain box? (1) If the crayons were evenly

Question

How many crayons are in a certain box?

(1) If the crayons were evenly distributed among a group of 7 children, 3 crayons would be left over.
(2) If the crayons were evenly distributed among a group of 4 children, 2 crayons would be left over.

Bunuel · Accepted Answer

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Hope it helps.

gmatophobia · Answer

Mathematical Solution

n → Number of crayons

Statement 1

(1) If the crayons were evenly distributed among a group of 7 children, 3 crayons would be left over.

n = 7k + 3 | k is a multipying factor

Case 1 : k = 1 ; n = 10 
  Case 2 : k = 2 ; n = 17

As we have more than one possible value for n, the statement alone is not sufficient. We can eliminate A and D.

Statement 2

(2) If the crayons were evenly distributed among a group of 4 children, 2 crayons would be left over.

n = 4m + 2 | m is a multipying factor

Case 1 : m = 1 ; n = 6 
  Case 2 : m = 2 ; n = 10

As we have more than one possible value for n, the statement alone is not sufficient. We can eliminate B.

Combined

n = 7k + 3

n = 4m + 2

The common representation of n =

n = LCM(7,4)z + first common term | z is the multiplying factor

n = 28z + 10

Case 1 : z = 1 ; n = 38 
  Case 2 : z = 2 ; n = 66

As we have more than one possible value for n, the statement combined is not sufficient.

Option E  .

Solution based on Reasoning

The solution will be a positive integer.

The statements individually are not sufficient on their own as it's very evident that we can have multiple possible solutions.

After the statements are combined,

from statement 1: n = 7k + 3; represents points on the number line which are 7 units apart. The first point is 3 (k=0). 
from statement 2: n = 4m + 2; represents points on the number line which are 4 units apart. The first point is 2 (m=0).

As the starting points of the two series are not the same and the interval between the two series is not the same, will have common members to both series.

Hence, we can have more than one number common to both sequences on the number line.

Option E

Archit3110 · Answer

target find how many crayons are there
#1
If the crayons were evenly distributed among a group of 7 children, 3 crayons would be left over.
with multiples of 7 , 3 extra crayons
10,17,24,31,38
insufficient
#2
 If the crayons were evenly distributed among a group of 4 children, 2 crayons would be left over
possible numbers multiples of 4
10,14,18,20,26,31
insufficient
from 1 &2
10,31
insufficient

OPTION E

JeffTargetTestPrep · Answer

We need to answer the question:

n = ? (# of crayons)

Statement One Alone:

=> If the crayons were evenly distributed among a group of 7 children, 3 crayons would be left over.

n = 7q + 3, where q is a non-negative integer.

n can be equal to 3, 10, 17, etc.

Statement one is not sufficient. Eliminate answer choices A and D.

Statement Two Alone:

=> If the crayons were evenly distributed among a group of 4 children, 2 crayons would be left over.

n = 4p + 2, where p is a non-negative integer.

n can be equal to 2, 6, 10, etc.

Statement two is not sufficient. Eliminate answer choice B.

Statements One and Two Together:

If n = 10, then both statements are true. Additionally, if n = 10 + LCM(7,4) = 10 + 28 = 38, then both statements remain true.

The two statements together are not sufficient.

Answer: E

ruis · Answer

Hello! I am conflicting with the following... In DS two statements cannot contradict each other.
With my line of reasoning (could be wrong) I understood:
Statement 1) tells us there is an odd number of crayons
Whereas Statement 2) tells us there is and even number of crayons
1) Leaves us with different possible outcomes as well as 2)
T) not possible. Hence, answer E.

Bunuel · Answer

Not correct.

From (1) x = 7q + 3 = 7q + odd. This will be odd when q is even, for example, consider q = 2, which gives x = 17, and even when q is odd, for example, consider q = 1, which gives x = 10.

From (2) x = 7p + 2 = 7p + even. This will be even when p is even, for example, consider p = 2, which gives x = 14, and odd when p is odd, for example, consider p = 1, which gives x = 9.

Also, on the GMAT, two data sufficiency statements always provide true information, and these statements never contradict each other or the stem. If the statements contradict each other, then, as per GMAT standards, the question is flawed, not that the answer is E.

ruis · Answer

Thank you for the explanation Bunuel! It really helps! I understood "evenly distributed" as spread equally but in even amounts (i.e. 2, 4, 6, 8.....). Big reading comprehension mistake

KarishmaB · Answer

We can see that it is a question on remainders - specifically, different divisors, different remainders.

Say there are n crayons in the box.

(1) If the crayons were evenly distributed among a group of 7 children, 3 crayons would be left over.

n = 7a + 3
We know there can be many values of a (number of crayons given to each child) to get many values of n.
Not Sufficient

(2) If the crayons were evenly distributed among a group of 4 children, 2 crayons would be left over.

n= 4b + 2
We know there can be many values of b (number of crayons given to each child) to get many values of n.
Not Sufficient

Using both, 
 n = 7a + 3
n= 4b + 2
We know that we can find the first solution by hit and trial and then get a general case. 
n = 28k + First solution

So n can still take infinite values. Should I find that solution? No, not required. I already have the answer.

Answer (E)

Goldenfuture · Answer

How many crayons are in a certain box?

(1) If the crayons were evenly distributed among a group of 7 children, 3 crayons would be left over.
7k + 3 = number
(2) If the crayons were evenly distributed among a group of 4 children, 2 crayons would be left over.
4q+2 = number

number is of form 28q+ 28 
So it can take multiple values and so answer is E

3,10,17,24, 31, 38, 
2,6,10,14, 18,22, 26,30, 34, 38, 42

ariiiianna · Answer

Why isn t 7n+3=4n+2?

ParamjitDasGMAT · Answer

Hi,  ariiiianna ,

This is because we don’t know if the values of ‘n’ are the same.

Indeed, assuming the same value of n would not work, as the equation would expand to:

3n = -1
Or, n= -1/3, which isn’t feasible.

Hope this helps.

ariiiianna · Answer

Hi, thank you very much!

How many crayons are in a certain box? (1) If the crayons were evenly

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How many crayons are in a certain box? (1) If the crayons were evenly

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