Question

Show the any positive odd integers is of the form 6q+1,6q+3,6q+5where is an integer

Answer 1

Answer :

Let "a" be any positive integer and b=6.

Then by Euclid's division lemma, there exists integers "a" and "r" such that :

a = bq + r, where 0 ≤ r < 6

a = 6q + r

=> a = 6q + 0 (when r = 0)

6q is even as 6 is divisible by 2 and any even number multiplied by an odd/even number will give even number.

=> a = 6q + 1 (when r = 1)

6q + 1 is odd as 6q is even but when we will add 1 it will become odd

=> a = 6q + 2 (when r = 2)

Even

=> a = 6q + 3 (when r = 3)

Odd

=> a = 6q + 4 (when r = 4)

Even

=> a = 6q + 5 (when r = 5)

Odd

So,

a = 6q+1 or, a = 6q+3 or, a = 6q+5 are odd

Hence, any positive odd integer is of the form 6q+1, 6q+3, 6q+5