Q4  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.

       [Hint : Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square
       each of these and show that they can be rewritten in the form 3m or 3m + 1.] 

       

Answers (1)

Let x be any positive integer.

It can be written in the form 3q + r where q\geq 0 and r = 0, 1 or 2

Case 1:

For r = 0 we have

x2 = (3q)2

x2 = 9q2

x2 = 3(3q2)

x2 = 3m

Case 2:

For r = 1 we have

x2 = (3q+1)2

x2 = 9q2 + 6q +1

x2 = 3(3q2 + 2q) + 1

x2 = 3m + 1

Case 3:

For r = 2 we have

x2 = (3q+2)2

x2 = 9q2 + 12q +4

x2 = 3(3q2 + 4q + 1) + 1

x2 = 3m + 1

Hence proved.

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