Question

Show that any positive odd integer is of the form 6q + 1 or, 6q + 3 or, 6q + 5, where q is some integer.

Answer 1

Let a be a given integer.

On dividing a by 6 , we get q as the quotient and r as the remainder such that

a = 6q + r, here r = 0,1,2,3,4,5
when r=0
a = 6q, even no
when r=1
a = 6q + 1, odd no
when r=2
a = 6q + 2, even no
when r = 3
a=6q + 3, odd no
when r=4
a=6q + 4 even no
when r=5,
a= 6q + 5 odd no

.Any positive odd integer is of the form 6q+1 ,6q+3 or 6q+5