Question

prove that if p and q are both positive integer then p square + q square is even but not divisible by 4​

Answer 1

Given:

p and q are both positive integer

To Prove:

p square + q square is even but not divisible by 4​

Solution:

We know any number can be written in the form aM + r , where a is any integer and r < a , is the reminder when the number is divided by a.

• Let  p = 2m + r1 , r1 ∈ {0,`1}
• and q = 2n + r2 , r2 ∈ {0,1}

Now lets find p²+q².

• p² + q² = (2m+r1)² + (2n+r2)²
• p² + q² = 4m² + 4mr1 + r1² + 4n² + 4nr2 + r2²
• p² + q² = 4(m² + n² + mr1 + mr2) + r1²+ r2²
• p²+q² = 4K + r1² + r2²

Since r1 can be 0 or 1, r1² can be 0 or 1.

r2 can also be 0 or 1 , r1² can be 0 or 1.

• r1² + r2² can be 0 , 1 , 2.
• Therefore,
• p² + q² can be 4K , 4K + 1 or 4K + 2.
• 4K + 2 is even .
• But none of the numbers are divisible by 4.

Hence proved that if p and q are both positive integer then p square + q square is even but not divisible by 4​