If x and y are both positive integers, x is a multiple of 3 and y is a

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.

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.

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

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.

Answer: B

Official Explanation:
In order for xy to a be a multiple of 75, it must have two 5’s and one 3 as prime factors. X has a 3, and Y has a 3 and a 7. This means that the statements must give us two 5’s in order to answer yes.
Statement 1 says that x is a multiple of 9. We now know x has two 3’s as prime factors, but we still don’t have any 5’s. Insufficient.
Statement 2 tells us that y is a multiple of 25. Since it is also a multiple of 21, we know that y has two 5’s (from 25) and a 3 (from 21) as prime factors, so xy must be a multiple of 75. Sufficient, so the correct answer is B.

ScottTargetTestPrep

Quick question:
Is the info in st.1 not enough to establish that xy is not a multiple of 75?

No, statement 1 alone does not provide sufficient information.

If x = 9*25 = 225 and y = 21, then all the conditions (x is a multiple of 3, x is a multiple of 9, y is a multiple of 21) are satisfied and in this case, xy = 4725 is a multiple of 75.

However, if x = 9 and y = 21, then again, all the conditions are satisfied, but in this case, xy = 189 is not a multiple of 75.

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