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Given below is a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statements is sufficient to answer the question. You should use the given data and your knowledge of Mathematics to choose between the possible answers. What will be the remainder left when MN is divided by 2, given M and N are natural numbers? I. M is not a multiple of 2. II. N is not a multiple of 2. 

Given below is a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statements is sufficient to answer the question. You should use the given data and your knowledge of Mathematics to choose between the possible answers. What will be the remainder left when MN is divided by 2, given M and N are natural numbers? I. M is not a multiple of 2. II. N is not a multiple of 2.  Correct Answer The data in statement I alone is sufficient to answer the question, while the data in statement II alone is not sufficient to answer the question.

From statement I:

M is not a multiple of 2. So, M is odd. It can be written as (2k + 1).

⇒ MN = (2k + 1)N

Using binomial theorem, if N is natural number, then all terms of expansion will be multiple of 2, except 1N. So, remainder will be 1.

∴ Statement I alone is sufficient to answer the question.

From statement II:

N is not a multiple of 2. M can be even or odd. Accordingly, MN can be even or odd, irrespective of value of N. And leave different remainders when divided by 2.

∴ Statement II alone is not sufficient to answer the question.

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