gulatin2 wrote:

If p, x, and y are positive integers, y is odd, and p = x^2 + y^2, is x divisible by 4?

(1) When p is divided by 8, the remainder is 5.

(2) x – y = 3

(1) If the remainder is 5 when p is divided by 8, x (or x^2) has to be an even integer since y is already said to be an odd integer. However its not sure whether x could be 2 or 4 or 6 or 8 or so on...

If x = 2, y could be any odd integer. Lets say y = 1, p = x^2 + y^2 = 5, which has 5 reminder when it is divided by 8.

If x = 2 and y = 3, p = x^2 + y^2 = 13, which has 5 reminder when it is divided by 8.

If x = 2, and y = 5, p = x^2 + y^2 = 29, which has 5 reminder when it is divided by 8.

But if x = 4, none of the values of y generates 5 reminder when p is divided by 8. For ex:

If x = 4, and y = 1, p = x^2 + y^2 = 17, which has 1 reminder when it is divided by 8.

If x = 4, and y = 3, p = x^2 + y^2 = 25, which has 1 reminder when it is divided by 8.

If x = 4, and y = 5, p = x^2 + y^2 = 41, which has 1 reminder when it is divided by 8.

If x = 8, and y = 1, p has 1 reminder when it is divided by 8.

If x = 8, and y = 3, p has 1 reminder when it is divided by 8.

If x = 8, and y = 5, p has 1 reminder when it is divided by 8.

Any value for x i.e. divisible by 4 doesnot produce 5 reminder when p is divided by 8.

So it is sufficient to answer that x is not divisible by 4.

(2) If x = y+3, p = x^2 + y^2 = (y+3)^2 + y^2. Here y could be 1, or 3 or 5 or 7 or so on....

If y = 1, x = 4...........yes.

If y = 3, x = 6...........no.

If y = 5, x = 8........... yes..

If y = 7, x = 10.............no.................NSF...

So that answer is A.

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