According to Euclid’s algorithm p = 6q + r

where r is any whole number 0< = r<6 and p is a positive integer

Since 6q is divisible by 2 so the value of r will decide whether it is odd or even.

Also since r<6 so only 6 cases are possible

For r = 1 , 3, 5 we get three odd numbers and for r = 0 , 2 , 4 we get three even numbers

So (6q + 1) , (6q + 3) & (6q + 5) represents positive odd integers .

Hence Proved