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# Is m divisible by n?

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Re: Is m divisible by n? [#permalink]
Bunuel wrote:
Is m divisible by n?

(1) mn/2 is divisible by n
(2) m + n is divisible by n.

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables (m and n) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
m + n = nk from the condition 2).
m = (k-1)n
Thus, both conditions together are not sufficient.

Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA

(Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
If m = 2, n = 2, the answer is 'yes'.
If m = 2, n = 3, the answer is 'no'.
Thus the condition 1) is not sufficient.

Condition 2)
m + n = nk from the condition 2).
m = (k-1)n
Thus, the condition 2) is sufficient.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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Re: Is m divisible by n? [#permalink]

Solution:

To find:

• If m is divisible by n or not.

o Inference: if m is divisible by n, m should be a multiple of n.

 This means, m can be represented in the form of: nx, where x is any positive integer.

Working out:

Statement 1:

mn/2 is divisible by n

This means, we can write mn/2 as a multiple of n

Translating the above statement in the mathematical form, we get:

$$mn/ 2 = nK$$, where K is a positive integer.

Dividing both the sides of this equation by n, we get:

$$m/2 = K$$

or, $$m =2K$$.

Since m is not of the form of nx, Statement 1 alone is not sufficient to answer this question.

Statement 2:

m + n is divisible by n

• From this statement, we can write that m+n = nK, where K is any positive integer.
Dividing both the sides of this equation by n, we get:

•$$m/n + 1 = K$$

Since K is an integer, and 1 is also an integer, m/n has to be an integer.

• Or, m/n can be written as y, where y is any integer.

• Thus, m/n = y, Or m = y*n

Since m can be represented in the form of n*a positive integer, m is divisible by n.

Hence, Statement 2 alone is sufficient to answer our question.

Re: Is m divisible by n? [#permalink]
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