Question

show that every even positive integer is of the form5m+3

Answer 1

let 'a' be even positive integer
By using Euclid division lemma
a=bq+r
r=1,2,3,4and5
5m ( r=0)
5m+1 (r=1)
5m+2 (r=2)
5m+3 (r=3)
5m+4 (r=4)
5m+5 (r=5)
the even positive integer is 5m, 5m+2and5m+4

Answer 2

let a be any positive integer and b = 5
then by Euclid division lemma,
a = bm + r
a = 5m + r
now r = 0, 1, 2, 3, 4
when r = 0, a = 5m
when r= 1, a = 5m + 1
when r = 2, a = 5m + 2
when r = 3, a = 5m + 3
when r = 4, a = 5m+ 4

and also every positive integer is of form 5m , 5m + 2, 5m + 4 .