Evenness of product property – UGC NET Question

This option is correct because the product of an even integer with any integer is always even. An even number can be written as 2k, and multiplying 2k by any integer m gives 2(km), which is still divisible by 2. Therefore, if at least one factor is even, the product is even.

Option A:
Even numbers are multiples of 2, and when such a number multiplies any integer, the factor of 2 remains in the result. This ensures the product is also a multiple of 2 and hence even. Thus, this option correctly states the property described.

Option B:
An odd product occurs only when both factors are odd, because then neither factor contributes a factor of 2. Here, the condition explicitly says at least one integer is even, so the product cannot be odd. Therefore, this option is wrong.

Option C:
A prime number has exactly two positive divisors, 1 and itself. The product of two integers with one even factor can be composite and need not be prime. So, prime is not guaranteed and is not the correct description.

Option D:
A negative product depends on the signs of the integers, not on whether they are even or odd. An even factor does not force the product to be negative. Hence, negativity is unrelated to the parity condition in the question.