Hi

(1) When the sum of A and B is divided by 5, the remainder is 1

A + B = 5x + 1. ------> A + B = 6, 11, 16 ... Putting this into remainders we can have (0 + 1), (1 + 0), (2 + 4), (4 + 2).

We are getting different values for A^2 - B^2 and remainders as a result. Insufficient.

(2) When B is subtracted from A and then divided by 3, the remainder is 1.

A - B = 3y + 1. Same logic as in previous case. Insufficient.

(1)&(2) Combining two:

\(A + B = 6, 11, 16, 21, 26, 31, 36 ....\)
\(A - B = 1, 4, 7, 10, 13, 16, 19, 22 ...\)

A + B = 11, A - B = 1 ----> A = 6, B = 5 ------->\(\frac{A^2 - B^2}{15} = \frac{36 - 25}{15} = \frac{16}{15}\) ----> remainder 1.

A + B = 16, A - B = 4 ----> A = 10, B = 6 -------> \(\frac{A^2 - B^2}{15} = \frac{100 - 36}{15} = \frac{64}{15}\) ----> remainder 4.

Insufficient

Another way to look at this:

\(A^2 - B^2 = (A - B)*(A + B) = (5x + 1)(3y+1) = 15xy + 5x + 3y + 1.\)

We can ingnore \(15xy\) which brings remainder \(0\). \(\frac{5x + 3y + 1}{15}\) will give us diferent remainders for different values of \(x\) and \(y\). Insufficient.

Answer E.