Question

Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q +1, where q is some integer.

Answer 1

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Answer 2

Let a be any positive integer and b = 2
By Euclid's theorem a = 2q + r where 0 is less than or equal to r and that is = b(2)
All possible values of r are 0 and 1
a=2q + 0 =2q
a=2q + 1
If a is in form 2q then it is even
A positive integer can be either odd or even
So we conclude that a positive odd integer is 2q+1