Question

Use division Algorithm to show that any positive odd integer is of the form 6q+1 or 6q+3 or 6q+5

Answer 1

let n be the positive odd integer
on dividing n by q let r be the remainder
by Euclid's division lemma n=6q+r, where r is 0,1,2,3,4,5
case 1, where r is 0
n=6q
case 2, where r is 1
n=6q+1
case 3, where r is 2
n=6q+2
case 4 where r is 3
n=6q+3
case 5 , where r is 4
n=6q+4
case 6, where r is 5
n=6q+5

clearly in case 2,4 and 6 it is odd
hence any +ve odd integer is in the form of 6q+1 , 6q+3 and 6q+5