To prove:
any positive odd integer is of the form 6q + 1 or, 6q + 3 or, 6q + 5,
where q is some integer.
Solution:
Let ‘a’ be any odd positive integer we need to prove that a is of the form 6q+1, or 6q+3, or 6q+5, where q is some integer.
Since a is an integer consider b = 6 another integer applying
Euclid's division lemma there exist integers q and r such that we get,
a = 6q + r for some integer q ≠ 0,
and r = 0, 1, 2, 3, 4, 5
since 0 ≤ r < 6.
Therefore according to question:
a = 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5
However since a is odd so ‘a’ cannot take the values 6q, 6q+2 and 6q+4
(since all these are even integers, hence divisible by 2)
Therefore a = 6q + 1, a = 6q + 3, a = 6q + 5
