Bunuel wrote:
If x and y are positive integers, what is the remainder when (x^2 + y^2) is divided by 4 ?
(1) x and y are consecutive integers
(2) x is even and y is odd
To know this, we need to know the remainder of x^2 and y^2 when divided by 4.
As equations look a bit complex and our statements are very simple, we'll first try a few numbers.
This is an Alternative approach.
(1) As there are only 4 possible classes of integers with respect to 'remainder when divided by 4' (remainder = 0,1,2,3), checking the first 4 pairs is enough.
Say x = 1, y = 2. Then 1^2+2^2 = 5 which has remainder 1. Say x = 2, y = 3. Then 4 + 9 = 13 which has remainder 1. Say x = 3, y = 4. Then 9+16=25 which also has remainder 1. Last x = 4, y = 5 Which gives 16+25=41, remainder 1.
Then (1) is sufficient.
(2) Once again, there are only 4 options, depending on the remainder classes when divided by 4: x has remainder 0 and y remainder 1, x has remainder 0 and y remainder 3, x remainder 2 and y remainder 1 and x remainder 2 and y remainder 3. Looking at (1), we'll see what we've already checked: (4,5) is the first option, (4,3) is the second option, (2,1) is the third option and (2,3) the fourth option. So we'll get the same answer.
Sufficient.
(D) is our answer.
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