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# Last Monday N female executives (N>1) received M male managers (M>1)

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GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
Last Monday N female executives (N>1) received M male managers (M>1)  [#permalink]

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26 Mar 2019, 07:32
00:00

Difficulty:

55% (hard)

Question Stats:

56% (02:18) correct 44% (02:32) wrong based on 50 sessions

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GMATH practice exercise (Quant Class 18)

Last Monday N female executives (N>1) received M male managers (M>1) for a business meeting. If every person shook hands exactly once with every other person in the meeting, what is the difference between the total number of shaking hands and the number of shaking hands among the female executives only?

(1) M < 11
(2) M(M+2N) = 65

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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Re: Last Monday N female executives (N>1) received M male managers (M>1)  [#permalink]

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26 Mar 2019, 08:10
The question is simply asking what are the values of m and n... (Both are integers)..

Statement 1 is clearly not sufficient as m and n can take numerous values...

Statement 2 -
M(m + 2n) = 65
Now factors of 65 include 1,5,13,65...

(1,65) are ruled out since both m and n are greater than 1..

M(m+2n) = 5*13
M = 5 , 5+2n = 13
N = 4... Bingo... That's all what we need.... (We cannot take m as 13 because taking m as 13 would give n a negative value).

Statement 2 individually is sufficient to answer the question..

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GMATH Teacher
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Last Monday N female executives (N>1) received M male managers (M>1)  [#permalink]

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26 Mar 2019, 14:39
fskilnik wrote:
GMATH practice exercise (Quant Class 18)

Last Monday N female executives (N>1) received M male managers (M>1) for a business meeting. If every person shook hands exactly once with every other person in the meeting, what is the difference between the total number of shaking hands and the number of shaking hands among the female executives only?

(1) M < 11
(2) M(M+2N) = 65

$$m,n\,\, \ge \,\,2\,\,\,{\rm{ints}}\,\,\,\,\left( * \right)$$

$$? = C\left( {m + n,2} \right) - C\left( {n,2} \right) = {{\left( {m + n} \right)\left( {m + n - 1} \right)} \over 2} - {{n\left( {n - 1} \right)} \over 2}$$

$$? = \frac{{m\left( {m + n - 1} \right) + nm + n\left( {n - 1} \right) - n\left( {n - 1} \right)}}{2} = \,\,\frac{{m\left( {m + 2n - 1} \right)}}{2}\,\,\,\,\, \Leftrightarrow \,\,\,\,\boxed{\,? = \frac{{m\left( {m + 2n - 1} \right)}}{2}\,}$$

$$\left( 1 \right)\,\,m < 11\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2,2} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{5}} \hfill \cr \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2,3} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\, \ne \,\,{\rm{5}}\, \hfill \cr} \right.$$

$$\left( 2 \right)\,\,m\left( {m + 2n} \right) = 65 = 5 \cdot 13\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)\,\,{\rm{and}}\,\,\left( {**} \right)} \,\,\,\,\left( {m,m + 2n} \right) = \left( {5,13} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$

$$\left( {**} \right)\,\,\,m > m + 2n\,\,\,\,\, \Rightarrow \,\,\,n < 0\,\,\,\,\,\,{\rm{impossible}}$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Last Monday N female executives (N>1) received M male managers (M>1)   [#permalink] 26 Mar 2019, 14:39
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