Question

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

Answer 1

Answer:

answer is in attachment......

Answer 2

↬Solution :-

Let a be a given positive odd integers

$\rm\:on\:dividing\:a\:by\:6,$

let q be the quotient and r be the remainder

⸕ By Euclid's algorithm,

We know that,

a = bq + r ( where 0 ≤ r < b )

so,

a = 6q + r ( where 0 ≤ r < 6 )

Possible values of r = 0,1,2,3,4,5

➝ a = 6q or a = 6q +1 or a = 6q +2 or a = 6q +3 or a = 6q + 4 or a = 6q +5

But a = 6q ,a = 6q +2 or a = 6q + 4 give even value of a

Hence, when a is odd it is of the form

➝ a = 6q + 1 or a = 6q + 3 or a = 6q + 5 some integers q.

Hence, every positive odd integers is of the form (6q + 1 ) or (6q + 3) or (6q + 5) for some integers q.