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# If m and n are positive integers and m*n = 40 what is m+n?

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Intern
Joined: 21 May 2016
Posts: 27
If m and n are positive integers and m*n = 40 what is m+n?  [#permalink]

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Updated on: 25 Sep 2018, 20:37
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55% (hard)

Question Stats:

57% (02:06) correct 43% (02:00) wrong based on 44 sessions

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If m and n are positive integers and m*n = 40 what is m+n?

(1) The number of positive factors of m is twice the number of positive factors of n.

(2) m has 4 different positive factors

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Originally posted by a70 on 25 Sep 2018, 09:55.
Last edited by Bunuel on 25 Sep 2018, 20:37, edited 2 times in total.
Manager
Joined: 02 Aug 2015
Posts: 153
Re: If m and n are positive integers and m*n = 40 what is m+n?  [#permalink]

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25 Sep 2018, 11:01
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a70 wrote:
If m and n are positive integers and m * n = 40 what is m=n?

1. The number of positive factors of m is twice the number of positive factors of n.

2. m has 4 different positive factors

Assuming the question asks for m and n,

m*n=40
m*n=2^3 * 5

Using statement 1, 2^3 has 4 factors and 5 has 2 factors, so m=2 and n=5. Sufficient.

Using statement 2, 2^3 has 4 different factors, so m=2 and n=5. Sufficient.

Hence D.

Cheers!
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
If m and n are positive integers and m*n = 40 what is m+n?  [#permalink]

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25 Sep 2018, 11:44
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1
a70 wrote:
If m and n are positive integers and m * n = 40 , what is the value of the sum of m and n?

(1) The number of positive factors of m is twice the number of positive factors of n.

(2) m has exactly 4 different positive factors

$$\left. \begin{gathered} m,n\,\, \geqslant 1\,\,\,{\text{ints}}\,\, \hfill \\ mn = 40\, \hfill \\ \end{gathered} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,m\,\,{\text{and}}\,\,n\,\,{\text{are}}\,\,{\text{pairs}}\,\,{\text{of}}\,{\text{positive}}\,\,{\text{factors}}\,\,{\text{of}}\,\,40$$

$$? = m + n$$

This is a perfect opportunity to present our "T diagram" (see image attached), GMATH´s creation to help students find ALL positive factors of a given positive integer.

We all know that 40 = (2^3)*5 has 4*2 = 8 positive factors. The T technique shows them explicitly and, more than that, in the corresponding pairs (in the four rows)!

We have put the number of positive factors of each positive factor of 40 in parentheses. A quick inspection in each pair of divisors shows us that:

$$\left( 1 \right)\,\,\,\, \Rightarrow \,\,\,\left( {m,n} \right) = \left( {{2^3},5} \right)\,\,\,\,\, \Rightarrow \,\,\,? = 13\,\,\,\,\, \Rightarrow \,\,\,{\text{SUFF}}.\,\,\,\,\,\,$$

$$\left( 2 \right)\,\,\, \Rightarrow \,\,\,\left\{ \begin{gathered} \,\left( {m,n} \right) = \left( {2 \cdot 5,{2^2}} \right)\,\,\,\,\, \Rightarrow \,\,\,? = 14\,\,\,\,\, \hfill \\ \,\left( {m,n} \right) = \left( {{2^3},5} \right)\,\,\,\,\, \Rightarrow \,\,\,? = 13\,\,\,\, \hfill \\ \end{gathered} \right. \Rightarrow \,\,\,{\text{INSUFF}}.$$

The correct answer is therefore (A).

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

Regards,
Fabio.
Attachments

GMATH_T_diagram_25Set18.gif [ 14.9 KiB | Viewed 921 times ]

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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If m and n are positive integers and m*n = 40 what is m+n?   [#permalink] 25 Sep 2018, 11:44
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