Use Euclid’s lemma to show that square of any positive integer is of form 4m or 4m + 1 for some integer m. [CBSE-2011-560017, 560037; 2010-1040109-A1, 1040119-A2]
Sol. Let a be any integer. Let q be the quotient and r be the remainder, when a is divided by 4.
By Euclid’s division algorithm, we have : a = 4q + r, where r = 0, 1, 2, 3
For r = 0, a = 4q and a2 = 16q2 = 4(4q2), which is of the form 4m, where m = 4q2.
For r = 1, a = 4q + 1 and a2 = 16q2 + 8q + 1 = 4(4q2 + 2q) + 1, which is of the form 4m + 1, where m = 4q2 + 2q.
For r = 2, a = 4q + 2 and a2 = 16q2 + 16q + 4 = 4(4q2 + 4q + 1), which is of the form 4m, where m = 4q2 + 4q + 1.
For r = 3, a = 4q + 3 and a2 = 16q2 + 24q + 9 = 4(4q2 + 6q + 2) + 1, which is of the form 4m + 1, where m = 4q2 + 6q + 2.
Thus, a2 is either of the form 4m or 4m + 1 for some integer m.
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