Question

Show that every positive even integer is of the form 2m and that every positive odd integer is of the form 2m 1 where m is some integer

Answer 1

let 'a' be any positive integer and b=2
By euclid's division algorithm
a=bq+r 0≤ra=2q+r 0≤r<2
(i.e) r =0,1
r=0 , a=2q+0=> a=2q
r=1, a=2q+1
if a is the form of 2m then 'a' is an even integer and positive odd integer is of the form 2m+1

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Answer 2

a=bq+r
a=2m+0(r can be 0 1)
a=2m as m is multiplied by2 so it is even
again a=2m+1
so a is an odd no.