Show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q. Share with your friends Share 0 Saksham Tolani answered this Dear Student, Let a be the positive integer and b = 4. Then, by Euclid’s algorithm, a = 4m + r for some integer m ≥ 0 and r = 0, 1, 2, 3 because 0 ≤ r < 4. So, a = 4m or 4m + 1 or 4m + 2 or 4m + 3. So, (4m)2 = 16m2 = 4(4m2) = 4q, where q is some integer. (4m + 1)2 = 16m2+ 8m + 1 = 4(4m2 + 2m) + 1 = 4q + 1, where q is some integer. (4m + 2)2 = 16m2 + 16m + 4 = 4(4m2 + 4m + 1) = 4q, where q is some integer. (4m + 3)2 = 16m2 + 24m + 9 = 4(4m2 + 6m + 2) + 1 = 4q + 1, where q is some integer. Hence, The square of any positive integer is either of the form 4q or 4q + 1, where q is some integer. Regards! 0 View Full Answer
