For some integer q, every odd is of the form

For some integer q , every odd integer is of the form

For some integer m , every even integer is of the form

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]

Show that the square of any positive integer is either of the form 4q or 4q + 1 form some integer q

Prove that if a positive integer is of the form 6q+5, then it is of the form 3q+2 for some integer q, but not conversely.

If p is an integers, then every square integer is of the form

Show that the square of any positive integer is either of the form "4m" or "4m+1" for some integer "m.

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

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.