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09 Nov 2021, 03:51
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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
(1) \(x\) and \(y\) are consecutive integers
(2) \(x\) is even and \(y\) is odd
09 Nov 2021, 03:51
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Official Solution:
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
The above means that one is even and another is odd. An even number (whichever it is, \(x\) or \(y\)) when squared is a multiple of 4, so we need to find the remainder when an odd integer squared is divided by 4.
\(odd^2=(2k+1)^2=4k^2+4k+1= 4(k^2+k)+1\). This gives the remainder of 1 upon division by 4. Sufficient.
(2) \(x\) is even and \(y\) is odd. This one is basically the same as the first statement, so sufficient.
Answer: D
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
The above means that one is even and another is odd. An even number (whichever it is, \(x\) or \(y\)) when squared is a multiple of 4, so we need to find the remainder when an odd integer squared is divided by 4.
\(odd^2=(2k+1)^2=4k^2+4k+1= 4(k^2+k)+1\). This gives the remainder of 1 upon division by 4. Sufficient.
(2) \(x\) is even and \(y\) is odd. This one is basically the same as the first statement, so sufficient.
Answer: D
