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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.

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.

Grade:

1 Answers

Latika Leekha
askIITians Faculty 165 Points
7 years ago
Let ‘a’ be a positive integer. now, we using the Euclid’s lemma we need to prove that the square of a is either of the form 3m, 3m+1 or 3m + 2.
Now, a = 3l + s, for some integer l \geq 0, where s = 0, 1 or 2.
Therefore, a = 3l or 3l + 1 or 3l + 2.
Hence, a2 = (3l)2 or (3l + 1)2 or (3l + 2)2
= 9l2 , 9l2 + 6l + 1 or 9l2 + 12l + 4
= 3.3l2 or 3(3l2 + 2l) + 1 or 3(3l2 + 4l + 1) + 1
= 3.r1 or 3.r2 + 1 or 3.r3 + 1, where r1, r2, r3 are positive integers.
Hence, the square of a positive integer is either of the form 3m or 3m + 1.
Thanks & Regards
Latika Leekha
askIITians Faculty

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