Statement 2) The remainder when any integer is divided by m is an even number
Case 1 : any integer = 5 ; m = 3 Remainder = 2 (Even)
Case 2 : any integer = 5 ; m = 1 Remainder = 0 (Even)
=> Two different values of m

Could you guys please tell me where did I made the mistake?

You tested 2 different values for m, but not 2 different values for ANY integer. You should ask, "with this particular value for m, will I ALWAYS get an even remainder?"

If m = 3 and "any integer" = 5, then the remainder is 2 (even).
If m = 3 and "any integer" = 4, then the remainder is 1 (odd).

Thus, m = 3 does not keep our statement true. It's an invalid value.

So, rather than continuing to test numbers, just think: if m is a positive integer, what integer would ALWAYS result in an even remainder, whether divided into an even or an odd integer? The only option --> 1. 1 is a factor of every positive integer, so it will yield a remainder of 0 (even) when divided into any integer.

This is sufficient to tell us that m = 1.

manik11 wrote:What is the value of positive integer m?

(1) The remainder when any integer is divided by m is less than 5.
(2) The remainder when any integer is divided by m is an even number.

Target question: What is the value of positive integer m?

Statement 1: The remainder when any integer is divided by m is less than 5.

Useful rule concerning remainders: When positive integer N is divided by positive integer D, the remainder R is such that 0 < R < D
For example, if we divide some positive integer by 7, the remainder will be 6, 5, 4, 3, 2, 1, or 0

Given the above rule, statement 1 basically tells us that m is less than or equal to 6.
So, m could equal 6, 5, 4, 3, 2 or 1
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The remainder when any integer is divided by m is an even number.
How is this possible?
If we're not sure, let's IGNORE statement 2 and see what happens with different values of m.
m = 5. If we divide some positive integer by 5, the remainder will be 4, 3, 2, 1, or 0 (some odd and some even)
m = 4. If we divide some positive integer by 4, the remainder will be 3, 2, 1, or 0 (some odd and some even)
m = 3. Here, the remainder will be 2, 1, or 0 (some odd and some even)
m = 2. Here, the remainder will be 1, or 0 (one odd and one even)
m = 1. The remainder will be 0 (EVEN) AHA!!!
This is the only case where the remainder MUST BE EVEN.
So, it must be true that m = 1

Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent