Question

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

Answer 1

Let a be the positive odd integer which when divided by 6 gives q as quotient and r as remainder.

According to Euclid’s division lemma

$a = bq + r$

$b = 6$

Where,

$(0 ≤ r < 6)$

So,

$r = 0,1,2,3,4,5$

Case 1:

$If \: r = 1, \: then$

$a = 6q + 1$

The Above equation will be always as an odd integer.

Case 2:

$If \: r=3, \: then$

$a = 6q + 3$

The Above equation will be always as an odd integer.

Case 3:

$If \: r=5, \: then$

$a = 6q + 5$

The Above equation will be always as an odd integer.

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

Hence proved.