Question

Class 10th____________________Prove that if x and y are odd positive integers, then x²+y² is even but not divisible by 4.
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Answer 1

Prove that if x and y are odd positive integers, then x²+y² is even but not divisible by 4.

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Here is your perfect answer!

Let x be 2a+1 & y be 2b+1, (odd positive integers).

x² + y²

= (2a+1)² + (2b+1)²

= (2a)² + 2(2a)(1) + 1² + (2b)² + 2(2b)1 + 1²

= 4a² + 4a + 1 + 4b² + 4b + 1

= 4(a² + a + b² + b) + 2

= 4q + 2,

where q = a² + a + b² + b,

Hence it is even but not divisible by 4 as it leaves 2 when divided by 4.

Answer 2

$\huge \bf \red{ \mid{ \overline{ \underline{ANSWER}}} \mid}$


$\bf{QUESTION}$
Prove that if x and y are odd positive integers, then x²+y² is even but not divisible by 4.



$\bf{step \: by \: step \: explanation}$


we know that any odd positive integer is of the form 2q+1 for some integer q.

so, let x = 2m+1 and y = 2n+1 for some integers m and n.

•°• x²+y²= ( 2m+1)²+(2n+1)²


==> x²+y²=4(m²+n²)+4(m+n)+2


==> x²+y²= 4{(m²+n²)+(m+n)} +2


==> x²+y² = 4q +2 , where q= (m²+n²)+(m+n)

==> x²+y² is even and leaves remainder 2 when divided by 4.

==> x²+y² is even but not divisible by 4.


HENCE PROVED✔✔



$\huge \pink{ \boxed{ \boxed{ \mathbb{THANKS}}}}$