Remember that: Even * Even = Even Even * Odd = Even Odd * Odd = Odd

The products of any two integers would be odd, only if both the integers are odd. Similarly, the products of any two integers would be even if only one integer is even.

In this case, we don’t need to look for the nature of the other integer.

Statement (1)

Statement (1) states that a*b is an odd integer. It means that a and b both should be odd integers. But we don’t have any information regarding c; therefore Statement (1) alone is insufficient.

The correct answer would be among B/C/E.

Statement (2)

Statement (2) states that b*c is an odd integer, which again tells us that b and c both are odd integers. But we don’t have any information regarding a; therefore Statement (2) alone is insufficient.

The correct answer would be between either C or E.

Statements (1 & 2) together

Both statements together confirm that a, b and c are odd integers; therefore a*b*c*must be an odd integer.

Correct answer is C.