Question

Choose any two odd numbers. Is their product an odd number? Is it true for any two, even whole numbers?

Answer 1

Answer:

1

Step-by-step explanation:

Let’s assume a is an odd number. Odd and even numbers alternate and so a−1 must therefore be an even number and thus, by definition a−1 is divisible by 2. This means that a−1 is 2 times some integer n which we write as a−1=2n and this implies that

a=2n+1 for some integer n.

Let’s assume also that b is an odd number. By the same argument we have

b=2m+1 for some integer m.

Now let’s look at a×b :

ab = (2n+1)(2m+1)

= 4nm+2n+2m+1

= 2(2nm+n+m)+1

By definition 2(2nm+n+1) is even because it is divisible by 2 and therefore, becasue odd and even numbers alternate, ab=2(2nm+n+1)+1 must be odd.