Prove that the product of two odd numbers is odd.

Let x and y be two odd numbers.

Then x = 2k + 1 for some natural number k and y = 2l + 1 for some natural number l.

Multipying both x and y, we get :

⇒ xy = (2k + 1)(2l + 1)

⇒ xy = 2k(2l + 1) + 1(2l + 1)

⇒ xy = 4kl + 2k + 2l + 1

⇒ xy = 2(kl + k + l) + 1

Let (kl + l + 1) be M.

⇒ xy = 2M + 1

We know that,

Any natural number on multiplying by 2 is an even number.

On adding 1 to an even number we an odd number.

So, product of two odd numbers is odd.

Hence, proved that the product of two odd numbers is odd.