OE


We know that p is the remainder on dividing X by 2, while q is the remainder on dividing Y by 2; we need to check that which of the given statements is sufficient to find the relation between p and q, whether p is equal to q or greater or less.

As Dividend = Divisor x Quotient + Remainder , we get \(X = 2k_1 + p\) & \(Y = 2k_2 + q\) , where \(k_1\) & \(k_2\) are non negative integers i.e. X & Y are p & q greater, respectively, than the multiple of 2.

Note: - Since p & q are the remainders when X & Y are divided by 2, their values can be 0 or 1 only.

Now checking from the options we get

(A) X is an even integer — If X is an even integer the value of p must be 0 but as nothing is said about the value of Y we cannot say anything about the value of q, so the statement (A) is insufficient.

(B) Y is an odd integer - If Y is an odd integer the value of q must be an odd number i.e. 1 but as nothing is said about the value of X we cannot say anything about the value of p, so the statement (B) is insufficient.

(C) Y is a non-negative number — It gives nothing about the value of q or p, so the statement (C) is insufficient.

(D) X + Y + XY is an odd integer and q is not 0 — The expression X + Y + XY can be odd if exactly one of X and Y is even and the other one is odd or X & Y both are odd. If we consider exactly one of X & Y odd, as q is not 0, we must have X even & p = 0. So, the value of Y must be odd which gives q = 1. Thus, we get p < q. But if we take both X & Y odd, we must have p = q = 1. As different assumptions give different relations between p & q, statement (D) is insufficient.

(E) X + Y + XY is an even integer — which implies both X and Y are even, so the values of both p and q must be zero each. Hence, we get p = q = 0, so statement (E) is sufficient.

Hence only option (E) is correct.