Question

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

Answer 1

let a=some integer

b=2.

then, by Euclid's algorithm, a=2q+r,for some integer q≥0 and r=0 or r=1,because 0≤r<2.

let r=0, q=q, then

a=2q+r =

a=2q+0 =a=2q

let r=1

then, a=2q+r

a=2q+1,

so a=2q or 2q +1