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

Question

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

(2) m has 4 different positive factors

Diwakar003 · Answer

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!

fskilnik · Answer

\left. \begin{gathered}
 m,n\,\, \geqslant 1\,\,\,{	ext{ints}}\,\, \hfill \
 mn = 40\, \hfill \ 
\end{gathered} ight\}\,\,\,\,\, \Rightarrow \,\,\,\,m\,\,{	ext{and}}\,\,n\,\,{	ext{are}}\,\,{	ext{pairs}}\,\,{	ext{of}}\,{	ext{positive}}\,\,{	ext{factors}}\,\,{	ext{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 ight)\,\,\,\, \Rightarrow \,\,\,\left( {m,n} ight) = \left( {{2^3},5} ight)\,\,\,\,\, \Rightarrow \,\,\,? = 13\,\,\,\,\, \Rightarrow \,\,\,{	ext{SUFF}}.\,\,\,\,\,\,

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

The correct answer is therefore (A).

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

Regards, 
Fabio.

HoneyLemon · Answer

Given that m×n=40 where m and n are positive integers.

1. The number of positive factors of m is twice the number of positive factors of n
 
The positive factors of 40 are: 1, 2, 4, 5, 8, 10, 20 and 40

And the number of positive factors of these factors of 40 is as under:

1 has only 1 positive factor

2 has two positive factors

4 has three positive factors

5 has two positive factors

8 has four positive factors

10 has four positive factors

20 has six positive factors

40 has eight positive factors

Since, the number of positive factors of m is twice the number of positive factors of n and

as, m×n=40
Thus, the only possible values m and n could hold will be when;

n = 5 and m = 8 because 5 has two positive factors and 8 has four positive factors and their

product is 40

Therefore, m+n=5+8=13 hence giving us a definite value.

Hence, statement (1) alone is  sufficient .

2. m has four positive factors.
 
Since, m has four positive factors, so m could be 8 or 10

Accordingly n will be 5 or 4

Thus, m + n could be equal to (8 + 5) = 13 or (10 + 4) = 14, hence, not getting a definite value

of m + n.

Hence, statement (2) alone is  insufficient .
 answer is  A.

bumpbot · Answer

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

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