Q8 of 124 Page 2

Prove that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.

Since any positive integer n is of the form 2p or, 2p + 1

When n = 2p, then n2 = 4p2 = 4a where a = p2


When n = 2p + 1, then n2 = (2p + 1)2 = 4p2 + 4p + 1 ⇒ 4p(p + 1) + 1


⇒ 4m + 1 where m = p(p + 1)


Therefore square of any positive integer is of the form 4q or 4q + 1 for some integer q


More from this chapter

All 124 →
6

Prove that the square of any positive integer of the form 5q + 1 is of the same form.

 7

Prove that the square of any positive integer is of the form 3m or, 3m + 1 but not of the form 3m + 2.

 9

Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for some integer q.

 10

Show that the square of an odd positive integer is of the form 8q + 1, for some integer q.