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

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

Solution:

To Show: That any positive odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5 where q is any some integer.

Proof: Let a be any odd positive integer and = 6.

Then, there exists integers q and r such that 

$a=6 q+r, 0 \leq r<6$ (by division algorithm)

$\Rightarrow a=6 q$ or $6 q+1$ or $6 q+2$ or $6 q+3$ or $6 q+4$

But 6q or 6q + 2 or 6q + 4 are even positive integers.

So, $a=6 q+1$ or $6 q+3$ or $6 q+5$

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

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