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A school has a students and b teachers

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A school has a students and b teachers  [#permalink]

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New post 09 Sep 2018, 08:37
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Question Stats:

25% (02:32) correct 75% (02:38) wrong based on 52 sessions

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A school has a students and b teachers. If a < 150, b < 25, and classes have a maximum of 15 students, can the a students
be distributed among the b teachers so that each class has the same number of students? (Assume that any student can be
taught by any teacher)

(1) It is possible to divide the students evenly into groups of 2, 3, 5, 6, 9, 10, or 15

(2) The greatest common factor of a and b is 10
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A school has a students and b teachers  [#permalink]

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New post Updated on: 09 Sep 2018, 09:44
1
alphak3nny001 wrote:
A school has a students and b teachers. If a < 150, b < 25, and classes have a maximum of 15 students, can the a students
be distributed among the b teachers so that each class has the same number of students? (Assume that any student can be
taught by any teacher)

(1) It is possible to divide the students evenly into groups of 2, 3, 5, 6, 9, 10, or 15

(2) The greatest common factor of a and b is 10


Here is my take on this .

(1) It is possible to divide the students evenly into groups of 2, 3, 5, 6, 9, 10, or 15
we have to find LCM of the given numbers
We therefore have 90 students

case 1
say number of teachers = 15 (b<25)
then yes students can be evenly distributed

case 2
when no of teachers = 24
then no

(2) The greatest common factor of a and b is 10
a and b can be 100 and 10
or
\(90 and 20\)
or
90 and 10

insufficient

using both still insufficient

E

Originally posted by CounterSniper on 09 Sep 2018, 09:01.
Last edited by CounterSniper on 09 Sep 2018, 09:44, edited 1 time in total.
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Re: A school has a students and b teachers  [#permalink]

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New post 09 Sep 2018, 11:13
alphak3nny001 wrote:
A school has A students and B teachers. If A < 150, B < 25, and classes have a maximum of 15 students, can the A students be distributed among the B teachers so that each class has the same number of students? (Assume that any student can be taught by any teacher, any class has exactly one teacher associated with it.)

(1) It is possible to divide the students evenly into groups of 2, 3, 5, 6, 9, 10, or 15

(2) The greatest common factor of A and B is 10

\(2 \leqslant A \leqslant 149\,\,\,\operatorname{int} \,\,\,\,\left( * \right)\,\,\,\,\left( {A\,\, = \,\,\# \,\,{\text{students}}} \right)\)

\(2 \leqslant B \leqslant 24\,\,\,\operatorname{int} \,\,\,\,\,\,\,\,\,\left( {B\,\, = \,\,\# \,\,{\text{classes}}\,\,{\text{ = }}\,\,\,\# \,\,{\text{teachers}}} \right)\)

\(\left( {\frac{{n\,\,{\text{students}}}}{{1\,\,\,{\text{class}}}}} \right)\,\,\,\left( {B\,\,{\text{classes}}} \right)\,\,\,\mathop = \limits^? \,\,A\,\,\,\mathop \Leftrightarrow \limits^{1\,\, \leqslant \,\,n\,\,\operatorname{int} \,\, \leqslant \,\,15} \,\,\,\,\,\boxed{\,\,?\,\,\,\,:\,\,\,\,1 \leqslant \,\,\frac{A}{B}\,\,\, = \,\,\operatorname{int} \,\, \leqslant 15\,\,\,}\)

\(\left( 1 \right) \cap \left( * \right)\,\,\,\left\{ \begin{gathered}
A\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{multiple}}\,\,{\text{of}}\,\,LCM\left( {2,3,5,6,9,10,15} \right) = 90 \hfill \\
\left( * \right)\,\,\,\,2 \leqslant A \leqslant 149 \hfill \\
\end{gathered} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,A = 90\)

\(\left( {1 + 2} \right) \cap \left( * \right)\,\,\,\left\{ \begin{gathered}
A = 90\,\,\,\,\,;\,\,\,\,\,2 \leqslant B \leqslant 24\,\,\,\operatorname{int} \,\,\, \hfill \\
GCD\left( {A,B} \right) = 10 \hfill \\
\end{gathered} \right.\)

\(\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {{\text{A,B}}} \right) = \left( {90,10} \right)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\,\,\,\left( {\frac{{90}}{{10}} = 9} \right) \hfill \\
\,{\text{Take}}\,\,\left( {{\text{A,B}}} \right) = \left( {90,20} \right)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\,\,\,\,\left( {\frac{{90}}{{20}} \ne \operatorname{int} } \right) \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{INSUFF}}.\)


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

Regards,
fskilnik.
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Re: A school has a students and b teachers  [#permalink]

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New post 09 Sep 2018, 13:18
alphak3nny001 wrote:
A school has a students and b teachers. If a < 150, b < 25, and classes have a maximum of 15 students, can the a students
be distributed among the b teachers so that each class has the same number of students? (Assume that any student can be
taught by any teacher)

(1) It is possible to divide the students evenly into groups of 2, 3, 5, 6, 9, 10, or 15

(2) The greatest common factor of a and b is 10



The question asks if we can evenly divide the students into classes.

(1) It is possible to divide the students evenly into groups of 2, 3, 5, 6, 9, 10, or 15

The least common multiple =90 students

The number teachers < 25

Let b= 20

\(\frac{a}{b}=\frac{90}{20}=\frac{9}{2}\).............Answer is No

Let b= 10

\(\frac{a}{b}=\frac{90}{10}=\frac{9}{1}\).............Answer is yes

Insufficient

(2) The greatest common factor of a and b is 10

Use same examples above

Let a=90 & b=20

Let a=90 & b=10

Insufficient

Combine 1 & 2

Use same examples above........No conclusive answer

Insufficient


Answer: E
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Re: A school has a students and b teachers  [#permalink]

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New post 09 Sep 2018, 18:58
Mo2men wrote:
alphak3nny001 wrote:
A school has a students and b teachers. If a < 150, b < 25, and classes have a maximum of 15 students, can the a students
be distributed among the b teachers so that each class has the same number of students? (Assume that any student can be
taught by any teacher)

(1) It is possible to divide the students evenly into groups of 2, 3, 5, 6, 9, 10, or 15

(2) The greatest common factor of a and b is 10



The question asks if we can evenly divide the students into classes.

(1) It is possible to divide the students evenly into groups of 2, 3, 5, 6, 9, 10, or 15

The least common multiple =90 students

The number teachers < 25

Let b= 20

\(\frac{a}{b}=\frac{90}{20}=\frac{9}{2}\).............Answer is No

Let b= 10

\(\frac{a}{b}=\frac{90}{10}=\frac{9}{1}\).............Answer is yes

Insufficient

(2) The greatest common factor of a and b is 10

Use same examples above

Let a=90 & b=20

Let a=90 & b=10

Insufficient

Combine 1 & 2

Use same examples above........No conclusive answer

Insufficient


Answer: E


Hi,

I have a doubt. When we combine the statements. 90/20 is not a whole number. Hence we can eliminate it and 90/10 could be the solution. Please clarify.
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Re: A school has a students and b teachers &nbs [#permalink] 09 Sep 2018, 18:58
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