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

Question

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

blueseas · Answer

IMO A

126 = 9 * 7 * 2

NOW given
z= has atleast one 9
w has atleast one 4
now for zw to be multiple of 126 we need (9 * 7 * 2) so we are lacking only 7

statement 1: z is a multiple of 21
therefore z= 7*3*k
hence we have 7 too in zw THEREFORE YES.
SUFFICIENT.

statement 2:w is a multiple of 25
means we have two 5 's for sure but no idea about 7.
hence this is not sufficient.

hence A

mau5 · Answer

From F.S 1, we know that z is a multiple of both 9 and 21. Thus, z will be of the form z = 63*q(q is an integer). Now, zw will be of the form 63*4*k(an integer) = (63*2)*2k = 126*2k. Thus, zw is a multiple of 126. Sufficient.

From F.S 2, for w=0,z=0, we know that zw is a multiple of 126. However, for w=100,z=9, zw is not a multiple of 126. Insufficient.

A.

MacFauz · Answer

Question : 
z has at least two 3s, w has at least two 2s
Does zw have at least one 2, two 3s and one 7. i.e Do z or w have at least one 7 between them.

1) z has at least one 3 and one 7. Sufficient.
2) w has at least two 5s. Insufficient.

Answer is A

phferr1984 · Answer

I don't get this answer; I thought that zero is a multiple of every number and hence z and w could both be zero leading to answer E? The question stem doesn't seem to exclude z and w from being zero.

What am I missing?

Bunuel · Answer

But even if both z and w are 0, isn't zw = 0 a multiple of 126?

sanket1991 · Answer

z is a multiple of 9 and w is a multiple of 4, is zw a multiple of 126?
 Base work - zw = 126 = 3^2 * 2 * 7

A) z is a multiple of 21 and also 9 (given) so 
z is a multiple of 63*3
w is a multiple of 4 hence zw has to be a multiple 126
A sufficient

B) w is a multiple of 25 and also 4 (given) so
w is a multiple of 100
z is a multiple of 9
In this case zw is a multiple of 900 (no 7 factor) and hence zw of case B is not a multiple of 126

Answer : A only sufficient

amanvermagmat · Answer

Lets look at the prime factorisation of 126. 126 = 2 * 3^2 * 7. So for a number to be a multiple of 126, that number must have at least one 2, two 3's and one 7. So we are given that z has at least two 3's (multiple of 9) and w has at least two 2's (multiple of 4). So zw already has the required 2's and 3's. All we need to know is whether zw also has a 7 or not?

(1) z is a multiple of 21 = 3*7. So z has a 7, thus we know zw has a 7 too. Sufficient.

(2) w is a multiple of 25 = 5^2. So zw has two 5's also, but we dont know about 7. Insufficient.

Hence  A answer

fskilnik · Answer

\frac{z}{{{3^2}}} = \operatorname{int} \,\,\,\,\,;\,\,\,\,\,\,\,\frac{w}{{{2^2}}} = \operatorname{int}

\frac{{zw}}{{2 \cdot {3^2} \cdot 7}}\,\,\,\mathop = \limits^? \,\,\,\operatorname{int}

\left( 1 ight)\,\,\,\,\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} ight.\,\,\,\,\,\,\,\,\,\,

?\,\,\,:\,\,\,\,\frac{{zw}}{{2 \cdot {3^2} \cdot 7}} = \left( {\frac{z}{{{3^2} \cdot 7}}} ight) \cdot \left( {\frac{w}{2}} ight)\,\,\, = \,\,\,\operatorname{int} \,\, \cdot \,\,\,\operatorname{int} \,\,\,\, = \operatorname{int} \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{	ext{YES}}} ightangle

\left( 2 ight)\,\,\,\,\frac{w}{{{5^2}}} = \operatorname{int} \,\,\,\,\,\left\{ \begin{gathered}
 \,{	ext{Take}}\,\,\,\left( {z,w} ight) = \left( {0,0} ight)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{	ext{YES}}} ightangle \hfill \
 \,{	ext{Take}}\,\,\,\left( {z,w} ight) = \left( {{3^2},{2^2} \cdot {5^2}} ight)\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{{	ext{7}}\,\,{	ext{missing}}} \,\,\,\,\,\,\left\langle {{	ext{NO}}} ightangle \hfill \ 
\end{gathered} ight.\,\,\,\,\,

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

Regards, 
Fabio.

bumpbot · Answer

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

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

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