Show that any positive even int.eger is of the form 4q or 4q + 2,, where q is a whole number.

Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

Show that every positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

Show that any positive odd integer is of the form 6q+1, or 6q+3, or 6q+5, where q is some integer.

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

Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.

In each of the following questions, we ask you to prove a statement. List all the steps in each proof, ang give the reason for each step. 5. If a and b positive integers, then you know that a =bq +r, 0 £ r lt b , where q is a whole number. Prove that HCF (a,b) =HCF (b,r).

Show that the square of any positive odd integer is of the form 4q + 1 for any integer q.

Show that square of any odd integer is of the form 4q + 1 for some integer q.

Use Euclid's division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m . (Hint : Let x be any positive integer then it is of the form 3q , 3q +1 or 3q + 2 . Now square each of these and show that they can be rewritten in the form 3m or 3m + 1 .]