Prove that if x and y are odd positive integers then x² – y² divisible by 4.

We know that any odd positive integer is of the form

2p + 1;

where p = 0,1,2,…….

Therefore Let x = 2m + 1 and y = 2n + 1 for any integers m and n

Now, x² – y² = (2 m + 1)² – (2 n + 1)²

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

x² – y² = 4 m^{2 +} 4 m – 4 n² – 4 n

x² – y² = 4 (m² – n² + m – n)

x² – y² = 4 (m – n)(m + n + 1)

x² – y² = 4p; where p = (m – n)(m + n + 1)

Now when we divide x² – y² by 4 leaves no remainder

Hence, x² – y² is divisible by 4.