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Two teachers, Ms. Ames and Mr. Betancourt, each had N

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Two teachers, Ms. Ames and Mr. Betancourt, each had N

by BTGmoderatorLU » Wed Oct 31, 2018 2:59 pm

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Source: Magoosh

Two teachers, Ms. Ames and Mr. Betancourt, each had N cookies. Ms. Ames was able to give the same number of cookies to each one of her 24 students, with none left over. Mr. Betancourt also able to give the same number of cookies to each one of his 18 students, with none left over. What is the value of N?

1) N < 100
2) N > 50

The OA is A.
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Two teachers, Ms. Ames and Mr. Betancourt, each had N cookie

by fskilnik@GMATH » Wed Oct 31, 2018 6:22 pm

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BTGmoderatorLU wrote:Source: Magoosh

Two teachers, Ms. Ames and Mr. Betancourt, each had N cookies. Ms. Ames was able to give the same number of cookies to each one of her 24 students, with none left over. Mr. Betancourt also able to give the same number of cookies to each one of his 18 students, with none left over. What is the value of N?

1) N < 100
2) N > 50
$$\left\{ \matrix{
\left( {{\rm{Ames}}} \right)\,\,N = 24 \cdot Q\,\,,\,\,Q \ge 1\,\,{\mathop{\rm int}} \hfill \cr
\left( {{\rm{Betan}}} \right)\,\,N = 18 \cdot K\,\,,\,\,K \ge 1\,\,{\mathop{\rm int}} \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,N = LCM\left( {24,18} \right) \cdot J\,\,,\,\,J \ge 1\,\,{\mathop{\rm int}} $$

(N is a multiple of 18 and also a multiple of 24, hence N must be a multiple of the LCM of 18 and 24.)

$$LCM\left( {{2^3} \cdot 3,2 \cdot {3^2}} \right) = {2^3} \cdot {3^2} = 72\,\,\,\,\,\,\,\, \Rightarrow \,\,\,N = 72 \cdot J\,\,,\,\,J \ge 1\,\,{\mathop{\rm int}} $$
$$? = N$$
$$\left( 1 \right)\,\,N < 100\,\,\,\,\, \Rightarrow \,\,\,\,J = 1\,\,\,\, \Rightarrow \,\,\,\,? = 72$$
$$\left( 2 \right)\,\,N > 50\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,J = 1\,\,\,\, \Rightarrow \,\,\,\,? = 72 \hfill \cr
\,{\rm{Take}}\,\,J = 2\,\,\,\, \Rightarrow \,\,\,\,? = 144 \hfill \cr} \right.$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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by Jay@ManhattanReview » Wed Oct 31, 2018 11:14 pm

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BTGmoderatorLU wrote:Source: Magoosh

Two teachers, Ms. Ames and Mr. Betancourt, each had N cookies. Ms. Ames was able to give the same number of cookies to each one of her 24 students, with none left over. Mr. Betancourt also able to give the same number of cookies to each one of his 18 students, with none left over. What is the value of N?

1) N < 100
2) N > 50

The OA is A.
As per the information, we have N is a multiple of 24 and 18.

Say,

N/24 = p, where p is a positive integer; => N = 24p
N/18 = p, where q is a positive integer; => N = 18q

=> 24p = 18q

4p = 3q

=> p = 3q/4 => q must be a multiple of 4

q: {4, 8, 12, 16, 20, ...}

Thus, the corresponding values of N are: {72, 144, 288, 360, ...}

If we get the unique value of p or q, we get the answer.

Let's take each statement one by one.

1) N < 100

From the set for N: {72, 144, 288, 360, ...}, we have N = 72. Sufficient.

2) N > 50

From the set for N: {72, 144, 288, 360, ...}, we have {72, 144, 288, 360, ...}. No uniue value of N. Insufficient.

The correct answer: A

Hope this helps!

-Jay
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by GMATGuruNY » Thu Nov 01, 2018 3:14 am

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BTGmoderatorLU wrote:Source: Magoosh

Two teachers, Ms. Ames and Mr. Betancourt, each had N cookies. Ms. Ames was able to give the same number of cookies to each one of her 24 students, with none left over. Mr. Betancourt also able to give the same number of cookies to each one of his 18 students, with none left over. What is the value of N?

1) N < 100
2) N > 5.
Since the N cookies can be divided evenly among 24 or 18 students, N must be divisible by both 24 and 18.
One way to determine the LCM of two integers is to take multiples of the LARGER integer until we get a multiple of the SMALLER integer.
Multiples of 24:
24, 48, 72...
We can stop here: the value in blue is divisible by 18, since 72/18 = 4.
Thus, the LCM of 24 and 18 = 72.
Implication:
Since the N cookies can be divided evenly among 24 or 18 students, N must be a positive MULTIPLE OF 72.

Statement 1: N < 100
Thus, N=72.
SUFFICIENT.

Statement 2: N > 50
Here, N can be any positive multiple of 72.
INSUFFICIENT.

The correct answer is A.
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