. Use Euclid division lemma to show that the square of any positive integer is either of the form


3m or 3m + 1 for some integer m.

 

 

Best Answer

Solution:

By taking,’ a’ as  any positive integer and b = 3.

Applying Euclid’s  algorithm

a = 3q + r               

Here, r = remainder = 0,1,2 and q>0

So, a = 3q or 3q+1 or 3q+2

And,

a2 = (3q)2 or (3q+1)2 or (3q+2)2

a2 = (9q2)  or  9q2+6q+1 or  9q2+12q+4

a2 = 3(3q2)2 or (3q2 + 2q)+1 or 3(3q2+4q+1)+1

a2 = 3k1 or 3k2+1 or 3k3+1

Where k1, k2 and k3 are some positive integers

Hence, it can be said that the square of any positive integer is either of the form 3m or 3m+1.

 

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