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In the event where both x and y are odd positive integers, then is even but not divisible by.

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answer is A.

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Detailed Solution

According to Euclid’s division lemma,
Given positive integers say a and b, there exist unique integers q and r which satisfies

a = bq + r

, where

0 \leq r < b

.
Let’s consider the two odd positive numbers in the form,

x = 2 n + 1

and

y = 2 n + 3

then, on squaring both sides,

⇒ x 2 + y 2 = 2 n + 1 2 + 2 n + 3 2 ⇒ x 2 + y 2 = 4 n 2 + 4 n + 1 + 4 n 2 + 12 n + 9 ⇒ x 2 + y 2 = 8 n 2 + 16 n + 10 ⇒ x 2 + y 2 = 4 2 n 2 + 4 n + 2 + 2

Therefore, it shows that the above answer is even for every positive integer but not divisible by 4. Hence, it is verified that, if x and y are odd integers, then

x 2 + y 2

are even but not divisible by 4.
So, the correct option is 4.

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