Question

Show that every postive even integer is of the form 4q or 4q+2 and that every positive odd integer is of the form 4q+1 or 4q+3 where q is some integer.

Answer 1

let a be any positive integer.and b=4. then by euclid's algorithm,a=4q+r,for some integer q>=0 and r=0,r=1,r=2,r=3. because 0<=r<4
if a is of the form 4q, then a is an even integer . also a positive integer can be either even or odd. therefore any positive odd integer is of the form 4q+1,
if you substitute the R value in the euclid's algorithm.4q ,and 4q+2 are even integers.and
however since a is odd. a cannot be 4q and 4q+2