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Intern  Joined: 28 Apr 2014
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If x and y are both positive integers, x is a multiple of 3 and y is a  [#permalink]

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5 00:00

Difficulty:   25% (medium)

Question Stats: 78% (01:25) correct 22% (01:39) wrong based on 247 sessions

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If x and y are both positive integers, x is a multiple of 3 and y is a multiple of 21, is xy a multiple of 75?

1) x is a multiple of 9.
2) y is a multiple of 25.

Originally posted by zahraf on 15 Sep 2015, 18:36.
Last edited by ENGRTOMBA2018 on 15 Sep 2015, 18:57, edited 1 time in total.
Formatted the question and renamed the topic
Intern  Joined: 28 Apr 2014
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Re: If x and y are both positive integers  [#permalink]

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I think this question has problem, because we can answer the question without knowing the options,
and from option 1 we can answer the question: no , because X and Y are not factor of 5, so why the answer is B?
option 2 lead to yes answer, but we should consider sufficiency only.
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If x and y are both positive integers, x is a multiple of 3 and y is a  [#permalink]

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1
zahraf wrote:
I think this question has problem, because we can answer the question without knowing the options,
and from option 1 we can answer the question: no , because X and Y are not factor of 5, so why the answer is B?
option 2 lead to yes answer, but we should consider sufficiency only.

You are not correct to say that you can answer the question without looking at the statements.

Lets analyse the question:

Per the question statement: x=3p and y=21q=3*7*q, where p and q are positive integers. xy=3*21*pq=63pq, thus if pq=25n, then "yes" but if pq$$\neq$$ 25n then "no". Lets analyse the statements.

Per statement 1, x=9r=$$3^2$$*r, this still does not provide any information about r or y and thus is NOT sufficient. If r=25, then yes, xy will be a multiple of 75 but if r is not a multiple of 25, then xy will not be divisible by 75.

Per statement 2, y=25r=$$5^2$$r but per the question statement, y = 21q as well ---> y = 3*$$5^2$$*7*m (this comes from the fact that 21 and 25 do not share any prime factor and as such if y must be a multiple of both 21 and 25, then it should atleast have all the prime factors of 21 and 25.).

So, if y = 3*$$5^2$$*7*m = 75*7*m, we see that xy will be a multiple of 75 no matter what the value of x be. Thus this statement is sufficient.

B is thus the correct answer.
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Re: If x and y are both positive integers, x is a multiple of 3 and y is a  [#permalink]

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zahraf wrote:
If x and y are both positive integers, x is a multiple of 3 and y is a multiple of 21, is xy a multiple of 75?

1) x is a multiple of 9.
2) y is a multiple of 25.

x & y are positive

x ==> is a multiple of 3

y ==> is a multiple of 21 or we can write multiple of $$7 * 3$$

Question ==> is xy multiple of 75 or we can write, is multiple of $$5 * 3 * 5$$

From these statement as we can see from above for any number to be multiple of 75, we will need two 5's and one 3

Lets check the options:

1) x is a multiple of 9

As we know that x is a multiple of 9 we ill have $$3 * 3$$, however we are still missing two 5's and value of y can either have these two 5's or not have, hence we cannot say based on this equation that xy is multiple of 75

Hence, (1) ===== is NOT SUFFICIENT

2) y is a multiple of 25

From above we will know that y will now have: 5 * 5

Also we know that in xy, x is a multiple of 3, so now we have the minimum requirement being met for xy to be multiple of 75.

Hence, (2) ===== is SUFFICIENT

Hence, Answer is B
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Re: If x and y are both positive integers, x is a multiple of 3 and y is a  [#permalink]

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zahraf wrote:
If x and y are both positive integers, x is a multiple of 3 and y is a multiple of 21, is xy a multiple of 75?

1) x is a multiple of 9.
2) y is a multiple of 25.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

There are 2 variables and 0 equation.

Thus C is the answer most likely.

This question is a key question by VA method, since it is related to Integers. By CMT(Common Mistake Type) 4A, we should check the answer A or B too.

We can express as follwings.

$$x = 3a$$ and $$y = 21b$$ for some integers $$a$$ and $$b$$.
The question if $$xy$$ is a multiple of $$75$$ is equivalent to the question if $$ab$$ is a multiple of $$25$$ since $$xy = 3a \cdot 21b = 3^2 \cdot 7 \cdot ab$$ an $$75 = 3 \cdot 5^2$$.

Condition 1)
Since $$x = 3a$$ is a multiple of $$9$$, $$a$$ is a multiple of $$3$$. We can determine from this condition if $$ab$$ is a multiple of $$25$$ or not.
Thus this is not sufficient.

Condition 2)
Since $$y = 21b$$ is a multiple of $$25$$, $$b$$ must be a multiple of $$25$$. Thus $$ab$$ is also a multiple of $$25$$.
Thus this condition is sufficient.

Therefore, B is the Answer

Normally for cases where we need 2 more equations, such as original conditions with 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore C has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) together. Here, there is 70% chance that C is the answer, while E has 25% chance. These two are the key questions. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer according to DS definition, we solve the question assuming C would be our answer hence using 1) and 2) together. (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: If x and y are both positive integers, x is a multiple of 3 and y is a  [#permalink]

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zahraf wrote:
If x and y are both positive integers, x is a multiple of 3 and y is a multiple of 21, is xy a multiple of 75?

1) x is a multiple of 9.
2) y is a multiple of 25.

St 1
x= 3 x some integer k
X= 3 x 3 x some integer K - so we could have

18 x 21 which is not a multiple of 75

St 2

Y= 5 x 5 x some integer k
and
Y= 7 x 3 x some integer k

Find the LCM

LCM(25, 21) = 7 x 5 x 5 x 3- so essentially

Y= 7 x 5 x 5 x 3 x some integer k - so basically

X(7 x 5 x 5 x 3) Must be a multiple of 75 given that X is an integer

B
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Re: If x and y are both positive integers, x is a multiple of 3 and y is a  [#permalink]

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zahraf wrote:
If x and y are both positive integers, x is a multiple of 3 and y is a multiple of 21, is xy a multiple of 75?

1) x is a multiple of 9.
2) y is a multiple of 25.

We are given that x is a multiple of 3 and y is a multiple of 21. We need to determine whether xy is a multiple of 75. Since we see that xy is divisible by 3, we really need to determine whether xy is divisible by 25.

Statement One Alone:

x is a multiple of 9.

Since statement one does not tell us whether x or y is divisible by 25, we cannot determine whether xy is divisible by 75. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

y is a multiple of 25.

Since y is a multiple of 25, we see that xy must be a multiple of 25, and thus xy is divisible by 75.

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Re: If x and y are both positive integers, x is a multiple of 3 and y is a  [#permalink]

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_________________ Re: If x and y are both positive integers, x is a multiple of 3 and y is a   [#permalink] 02 Feb 2019, 23:51
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