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 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.

Use division algorithm to show that the square of any positive integer is of the form 3p or 3p + 1.

Use division algorithm to show that the cube of any positive integer is of the form 9 m, 9m + 1 or 9m + 8.

If q is some integer, then any positive odd integer is of the form :

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

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.

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 .]