 Hello and welcome to the session. Today I will help you with the following question. The question says show that any positive odd integer is of the form 6q plus 1 or 6q plus 3 or 6q plus 5 where q is some integer. First let's see what Euclid's division lemma is. Given positive integers a and b there exist unique integers and r equal to bq plus r where r is greater than equal to 0 and less than b. This is the key idea to be used for this question. Euclid's division lemma and algorithm are so closely interlinked that people often call Euclid's division lemma as Euclid's division algorithm also. Now let's move on to the solution. Let a be a odd positive integer. We apply division algorithm or you can say Euclid's division lemma with a and b equal to 6. Now since a is greater than equal to 0 and less than 6 the possible remainders are 0, 1, 2, 3, 4 and 5. That is a can be 6q or 6q plus 1 or 6q plus 2 or 6q plus 3 or 6q plus 4 or 6q plus 5 where q is the question. Now since a cannot be 6q or 6q plus 2 or 6q plus 4, since divisible by therefore any positive integer is of the form 2 plus 1 or 6q plus 3 or 6q plus 5 where q is some integer. So this completes the question. So hope you enjoyed the session. Have a good day.