Mascarfi wrote:

If r, a and b are positive integers, is r a product of two consecutive odd integers?

1) \(r = 4a^2 - b^2\)

2) b = 1

2 Consecutive odd integers can be of the form : [2a+1 , 2a-1] or [2a+1, 2a+3] , with a \(\in\) integer.

Now, statement 2 is clearly not sufficient.

Per statement 1, \(r = 4a^2 - b^2\) ---> r = (2a+b)(2a-b). Thus r can be equal to product of 2 consecutive odd integers only if b =1 .

But per this statement, b can be any value (=3,5,4, etc.). Thus r = (2a+4)(2a-4) (this would give r = even and thus given question will be "no") or r = (2a+1)(2a-1) (this will give r = odd and thus the given question will be "yes").

Thus statement 1 is not sufficient.

Combining statements 1 and 2, we see r = (2a+1)(2a-1) and thus r is a product of 2 consecutive odd integers. Hence we get a definite answer by combining the statements and hence C is the correct answer.