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Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for some integer q. |
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Answer» By Euclid’s division algorithm a = bm + r, where 0 ≤ r ≤ b Put b = 5 a = 5m + r, where 0 ≤ r ≤ 4 If r = 0, then a = 5m If r = 1, then a = 5m + 1 If r = 2, then a = 5m + 2 If r = 3, then a = 5m + 3 If r = 4, then a = 5m + 4 Now, (5)2 = 25m2 = 5(5m2) = 5q where q is some integer (5m + 1)2 = (5m)2 + 2(5m)(1) + (1)2 = 25m2 + 10m + 1 = 5(5m2 + 2m) + 1 = 5q + 1 where q is some integer (5m + 1)2 = (5m)2 + 2(5m)(1)(1)2 = 25m2 + 10m + 1 = 5(5m2 + 2m) + 1 = 5q + 1 where q is some integer = (5m + 2)2 = (5m)2 + 2(5m)(2) + (2)2 = 25m2 + 20m + 4 = 5(5m2 + 4m) + 4 = 5q + 4, where q is some integer = (5m + 3)2 = (5m)2 + 2(5m)(3) + (3)2 = 25m2 + 30m + 9 = 25m2 + 30m + 5 + 4 = 5(5m2 + 6m + 1) + 4 = 5q + 1, where q is some integer = (5m + 4)2 = (5m)2 + 2(5m)(4) + (4)2 = 25m2 + 40m + 16 = 25m2 + 40m + 15 + 1 = 5(5m2) + 2(5m)(4) + (4)2 = 5q + 1, where q is some integer Hence, the square of any positive integer is of the form 5q or 5q + 1, 5q + 4 for some integer q. |
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