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.