Question

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

Answer 1

Let a be any positive odd integer and b=6
So, a=bq +r,0>r>b
r=1,2,3,4,5or 6
a is not in the form of 6q,6q+2,6q+4or 6q+6 because this numbers are divided by 2
So, a is in the form of 6q+1,6q+3 or 6q+5

Answer 2

let a is any positive integer.
We can write it in the form a=bq+r
,where b=6

Then we know that r is always less than the b . So r could be 0 ,1,2,3,4,5
Eq formed are 6q +0,6q+1, 6q+2,6q+3,6q+4,6q+5
Since a is odd(given) and 6q ,6q+2and 6q + 4 are even so they cannot be equal.
So any positive odd integer is of the form 6q+1,
6q+3and 6q +5 .

# proved.
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