Question

use division algorithm to show that any positive odds integer is of the form of 6q+1 or 6q+3 or 6q+5​

Answer 1

★SolutiOn:-

Let n be the positive odd integer

on dividing n by q let r be the remainder

${\bf{\blue{By\: Euclid's\:division\: lemma}}}$

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 (+)ive odd integer is in the form of 6q+1 , 6q+3 and 6q+5.