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If z is a multiple of 9 and w is a multiple of 4, is zw a

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If z is a multiple of 9 and w is a multiple of 4, is zw a

by fskilnik@GMATH » Mon Sep 24, 2018 4:17 am

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BTGmoderatorDC wrote:If z is a multiple of 9 and w is a multiple of 4, is zw a multiple of 126?

(1) z is a multiple of 21
(2) w is a multiple of 25

Source: Veritas Prep
\[\frac{z}{{{3^2}}} = \operatorname{int} \,\,\,;\,\,\,\,\,\frac{w}{{{2^2}}} = \operatorname{int} \,\,\,\,\,\left( * \right)\]
\[\frac{{zw}}{{2 \cdot {3^2} \cdot 7}}\,\,\,\mathop = \limits^? \,\,\,\operatorname{int} \]
\[\left( 1 \right)\,\,\,\,\left\{ \begin{gathered}
\,\frac{z}{{3 \cdot 7}} = \operatorname{int} \,\,\,\, \cap \,\,\,\,\frac{z}{{{3^2}}} = \operatorname{int} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\frac{z}{{{3^2} \cdot 7}} = \operatorname{int} \, \hfill \\
\,\frac{w}{{{2^2}}} = \operatorname{int} \,\,\,\,\,\, \Rightarrow \,\,\,\,\frac{w}{2} = \operatorname{int} \, \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\]
\[\frac{{zw}}{{2 \cdot {3^2} \cdot 7}} = \left( {\frac{z}{{{3^2} \cdot 7}}} \right) \cdot \left( {\frac{w}{2}} \right)\,\,\, = \,\,\,\operatorname{int} \,\, \cdot \,\,\,\operatorname{int} \,\,\,\, = \operatorname{int} \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \]
\[\left( 2 \right)\,\,\,\,\frac{w}{{{5^2}}} = \operatorname{int} \,\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\,\left( {z,w} \right) = \left( {0,0} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\
\,{\text{Take}}\,\,\,\left( {z,w} \right) = \left( {{3^2},{2^2} \cdot {5^2}} \right)\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{7}}\,\,{\text{missing}}} \,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\end{gathered} \right.\,\,\,\,\,\]

The correct answer is therefore [spoiler]__(A)_____[/spoiler] .

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

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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by Jay@ManhattanReview » Mon Sep 24, 2018 9:18 pm

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BTGmoderatorDC wrote:If z is a multiple of 9 and w is a multiple of 4, is zw a multiple of 126?

(1) z is a multiple of 21
(2) w is a multiple of 25

OA A

Source: Veritas Prep
BTGmoderatorDC wrote:If z is a multiple of 9 and w is a multiple of 4, is zw a multiple of 126?

(1) z is a multiple of 21
(2) w is a multiple of 25

OA A

Source: Veritas Prep
Say z = 9p and w = 4q, where p and q are some integers

=> zw = 36pq

zw = 36r; r = pq

If zw a multiple of 126, zw = 126m, where m is some integer

=> 36r = 126m

m = 36r / 126 = 2r/7

Since m is an integer, r must be a multiple of 7. Only statement 1 can assure that r is a multiple of 7 since Statement 1 says that z is a multiple of 21 (= 3*7). Sufficient.

The correct answer: A

Hope this helps!

-Jay
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by ceilidh.erickson » Wed Sep 26, 2018 4:00 pm

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BTGmoderatorDC wrote:If z is a multiple of 9 and w is a multiple of 4, is zw a multiple of 126?

(1) z is a multiple of 21
(2) w is a multiple of 25

OA A

Source: Veritas Prep
First, REPHRASE the given information:

- if z is a multiple of 9, it contains two factors of 3.
- if w is a multiple of 4, it contains two factors of 2.
- the product zw must contain two factors of 2 and two factors of 3.

Now rephrase the question:
Is zw a multiple of 126 --> does zw contain all of the prime factors of 126?
Break down 126 using a factor tree:
Image
The prime factorization of 126 = (2)(3^2)(7)

So we're wondering: does zw contain (2)(3^2)(7)?
We already know that it contains at least one 2 and two 3's. So the only thing we're left wondering: does zw contain a factor of 7?

Target question: does either z or w contain a factor of 7?

(1) z is a multiple of 21

This tells us that z contains a 7. Sufficient.

(2) w is a multiple of 25

This tells us nothing about whether the product will be divisible by 7. Insufficient.

The answer is A.
Ceilidh Erickson
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Harvard Graduate School of Education
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by ceilidh.erickson » Wed Sep 26, 2018 4:15 pm

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Btw, this question is nearly identical to this OG question: https://www.beatthegmat.com/is-xy-a-mul ... tml#712134
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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