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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.

```
3 months ago Anand Kumar Pandey
2937 Points
```							Dear StudentLet x be any positive integer and y = 3.By Euclid’s division algorithm, then,x = 3q + r for some integer q≥0 and r = 0, 1, 2, as r ≥ 0 and r < 3.Therefore,x = 3q, 3q+1 and 3q+2Now as per the question given, by squaring both the sides,we get,x^2=(3q)^2=9q^2=3×3q^2Let 3q^2= mThereforex^2= 3m ..........................(1)x^2= (3q + 1)^2= (3q)^2+1^2+2×3q×1= 9q^2+ 1 +6q= 3(3q^2+2q) +1Substitute,3q2+2q = m, to get,x^2= 3m + 1 .................................. (2)x^2=(3q+2)^2=(3q)^2+2^2+2×3q×2=9q^2+4+12q=3(3q^2+4q+1)+1Again, substitute, 3q^2+4q+1 = m, to get,x^2= 3m + 1................................. (3)Hence, from equation 1, 2 and 3, we can say that, the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.Thanks
```
3 months ago
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