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
If z is a multiple of 9 and w is a multiple of 4, is zw a
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\[\frac{z}{{{3^2}}} = \operatorname{int} \,\,\,;\,\,\,\,\,\frac{w}{{{2^2}}} = \operatorname{int} \,\,\,\,\,\left( * \right)\]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
\[\frac{{zw}}{{2 \cdot {3^2} \cdot 7}}\,\,\,\mathop = \limits^? \,\,\,\operatorname{int} \]
\[\left( 1 \right)\,\,\,\,\left\{ \begin{gathered}
\,\frac{z}{{3 \cdot 7}} = \operatorname{int} \,\,\,\, \cap \,\,\,\,\frac{z}{{{3^2}}} = \operatorname{int} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\frac{z}{{{3^2} \cdot 7}} = \operatorname{int} \, \hfill \\
\,\frac{w}{{{2^2}}} = \operatorname{int} \,\,\,\,\,\, \Rightarrow \,\,\,\,\frac{w}{2} = \operatorname{int} \, \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\]
\[\frac{{zw}}{{2 \cdot {3^2} \cdot 7}} = \left( {\frac{z}{{{3^2} \cdot 7}}} \right) \cdot \left( {\frac{w}{2}} \right)\,\,\, = \,\,\,\operatorname{int} \,\, \cdot \,\,\,\operatorname{int} \,\,\,\, = \operatorname{int} \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \]
\[\left( 2 \right)\,\,\,\,\frac{w}{{{5^2}}} = \operatorname{int} \,\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\,\left( {z,w} \right) = \left( {0,0} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\
\,{\text{Take}}\,\,\,\left( {z,w} \right) = \left( {{3^2},{2^2} \cdot {5^2}} \right)\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{7}}\,\,{\text{missing}}} \,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\end{gathered} \right.\,\,\,\,\,\]
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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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 integersBTGmoderatorDC 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
=> 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!
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First, REPHRASE the given information: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
- 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:
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
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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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
EdM in Mind, Brain, and Education
Harvard Graduate School of Education