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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187
Prove that if x and y are odd positive integers, then x²+y² is even but not divisible by 4.
Good question,
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.
Answered by
205
Prove that if x and y are odd positive integers, then x²+y² is even but not divisible by 4.
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✔✔
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