Prove that The product of two odd natural numbers must be od

Prove that: The product of two odd natural numbers must be odd.

Solution

let n1, n2 be two natural numbers

=>

there exists two natural numbers k1,k2 such that

n1 = 2k1-1,

n2 = 2k2 -1

=>

n1n2 = (2k1-1)(2k2 - 1) = 4k1k2 -2k2 -2k1 +1

= 2[2k1k2 -k1-k2 +1] -1

= 2m-1, where m = 2k1k2-k1-k2 +1

=>

n1n2 is odd

thus proved