Stste that any odd positive integers is of the form 8q+1,8q+3,8q+5,8q+7 for some integer q.

Stste that any odd positive integers is of the form 8q+1,8q+3,8q+5,8q+

1.	Stste that any odd positive integers is of the form 8q+1,8q+3,8q+5,8q+7 for some integer q.
Answer» Let n be a positive odd integer.\xa0We need to show that n can be written in any one of\xa0the form of\xa08q+1, 8q+3, 8q+5 or 8q+7According to division algorithm,we can write any number ‘a’ in the forma = 8q + rwhere q is any integer and 0 <= r <= 7. So r can be 0, 1, 2, 3, 4, 5, 6 or 7.Thus, a can be\xa0written asa = 8qa = 8q+2a = 8q+3a = 8q+4a = 8q+5a = 8q+6a = 8q+7We need only odd numbers. Since 8q, 8q+2, 8q+4, and 8q+6 are divisible by 2, they are even numbers.So any odd integer can be written as any one of the remaining forms which are (8q+1, 8q+3, 8q+5 or 8q+7.).