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m is a multiple of 13. Is mn a multiple of 195?

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m is a multiple of 13. Is mn a multiple of 195?

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m is a multiple of 13. Is mn a multiple of 195?

(1) n has every factor that 45 has.

(2) m is divisible by 18.

OA A

Source: Princeton Review

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BTGmoderatorDC wrote:
m is a multiple of 13. Is mn a multiple of 195?

(1) n has every factor that 45 has.

(2) m is divisible by 18.
Source: Princeton Review
$$m = 13K,\,\,K\,\,{\mathop{\rm int}} \,\,\,\,\,\left( * \right)$$
$$\frac{{m \cdot n}}{{3 \cdot 5 \cdot 13}}\,\,\mathop = \limits^? \,\,\operatorname{int} \,\,\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\,\boxed{\,\,\frac{{m \cdot n}}{{3 \cdot 5}}\,\,\,\mathop = \limits^? \,\,\,\operatorname{int} \,\,}$$
$$\left( 1 \right)\,\,\,n\,\,{\rm{has}}\,\,{\rm{3}}\,\,{\rm{and}}\,\,{\rm{5}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
$$\left( 2 \right)\,\,\,{m \over {2 \cdot {3^2}}} = {\mathop{\rm int}} \,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2 \cdot {3^2} \cdot 13,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$


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

Regards,
Fabio.

_________________
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
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BTGmoderatorDC wrote:
m is a multiple of 13. Is mn a multiple of 195?

(1) n has every factor that 45 has.

(2) m is divisible by 18.

OA A

Source: Princeton Review
Given that m is a multiple of 13, we have m = 13k, where k is any integer.

If mn a multiple of 195, mn = 195p, where p is any integer.

mn = 195p => 13kn = 195p => n = 195p/13k = 15p/k.

=> n = 15p/k

Let's take each statement one by one.

(1) n has every factor that 45 has.

Since 15 is a factor 45, the answer to the question is Yes. Sufficient.

(2) m is divisible by 18.

=> m is divisible by 13*18.

Thus, mn = 195p

13*18k*n = 195p
n = (5/6)*(p/k)

If p/k is a multiple of 6, the answer is Yes, else no. No unique answer. Insufficient.

The correct answer: A

Hope this helps!

-Jay
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Manhattan Review

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