Question

use division algorithm to show that any positive odd integer is of the form 6q+1,or 6q+3or 6q+5

Answer 1

Let a be any positive integer and b=6.Then by Euclid's division algorithm a=6q +r,for some integer q is greater and is equal to 0,and r=0,1,2,3,4,5,because r is greater and equal to 0and 2 is greater than r.
so,
a=6q+r=6q+0=6q(which is a positive even integer)
a=6q+r=6q+1(which is a positive odd interger)
a=6q+r=6q+2(which is a positive even integer)
a=6q+r=6q+3(which is a positive odd integer)
a=6q+r=6q+4(which is a positive even integer)
a=6q+r=6q+5(which is a positive odd integer)

Hence any positive odd integer is of the form 6q+1,or6q+3or6q+5.