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

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

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New post Updated on: 25 Sep 2018, 20:37
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Question Stats:

45% (01:42) correct 55% (01:36) wrong based on 29 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.
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Re: If m and n are positive integers and m*n = 40 what is m+n?  [#permalink]

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New post 25 Sep 2018, 11:01
1
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.

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

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New post 25 Sep 2018, 11:44
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.
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GMATH_T_diagram_25Set18.gif
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If m and n are positive integers and m*n = 40 what is m+n? &nbs [#permalink] 25 Sep 2018, 11:44
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