Question

Show that any positive odd integer is of form (6m+1) or (6m+3) or (6m+5), where m is some integer.

Answer 1

Let 'a' be the given integer on dividing 'a' by 6, we get 'm' as the quotient and r as the remainder.
Now, a = 6m + r, r = 0, 1, 2, 3, 4, 5
When r = 0
a = 6m, even number
When r = 1
a = 6m + 1, odd number
When r = 2
a = 6m + 2, even number
When r = 3
a = 6m + 3, odd number
When r = 4
a = 6m + 4, even number
When r = 5
a = 6m + 5, odd number
$\therefore$ Any positive odd integer is of the form 6m + 1, 6m + 3 or 6m + 5.