Question

prove that if x and y are both odd positive integer then X2 + Y2 is even but not divisible by 4

Answer 1

$<b>$Any odd positive integer is in the form of 2q + 1 where q is some integer.

Let x = 2n + 1 and y = 2m + 1,

where m and n are integer.

Now, x²+ y²= (2n + 1)² + (2m + 1)²

=> x²+ y²= 4n²  + 4n + 1 + 4m² + 4m + 1

=> x²+ y² = 4(n² + m²  + n + m) + 2

=>x²+ y² = 4p + 2, where p = n²  + m² + n + m

=> Since 4p and 2 are even numbers, So 4p + 2 is an even number.

=> x²  + y²  is an even number and leaves the remainder when divided by 4

Thanks

Have a colossal day ahead

Be Brainly

Answer 2

Answer:Any odd positive integer is in the form of 2q + 1 where q is some integer.

Let x = 2n + 1 and y = 2m + 1,

where m and n are integer.

Now, x²+ y²= (2n + 1)² + (2m + 1)²

=> x²+ y²= 4n²  + 4n + 1 + 4m² + 4m + 1

=> x²+ y² = 4(n² + m²  + n + m) + 2

=>x²+ y² = 4p + 2, where p = n²  + m² + n + m

=> Since 4p and 2 are even numbers, So 4p + 2 is an even number.

=> x²  + y²  is an even number and leaves the remainder when divided by 4

Step-by-step explanation: