4. Use Euclid's division lemma to show that the square of any positive integer is either ofthe form 3m or 3m + 1 for some integer R.[Hint: Let x be any positive integer then it is of the form 37, 39 +1 or 39+2. Now squareeach of these and show that they can be rewritten in the form 3m or 3m +1.]

4. Use Euclid's division lemma to show that the square of any positive

1.	4. Use Euclid's division lemma to show that the square of any positive integer is either ofthe form 3m or 3m + 1 for some integer R.[Hint: Let x be any positive integer then it is of the form 37, 39 +1 or 39+2. Now squareeach of these and show that they can be rewritten in the form 3m or 3m +1.]
Answer» let ' a' be any positive integer and b = 3.we know, a = bq + r , 0< r< b.now, a = 3q + r , 0<r < 3.the possibilities of remainder = 0,1 or 2Case I - a = 3qa2= 9q2= 3 x ( 3q2) = 3m (where m = 3q2)Case II - a = 3q +1a2= ( 3q +1 )2 = 9q2+ 6q +1 = 3 (3q2+2q ) + 1 = 3m +1 (where m = 3q2+ 2q )Case III - a = 3q + 2a2= (3q +2 )2 = 9q2+ 12q + 4 = 9q2+12q + 3 + 1 = 3 (3q2+ 4q + 1 ) + 1 = 3m + 1 where m = 3q2+ 4q + 1) From all the above cases it is clear that square of any positive integer ( as in this case a2) is either of the form 3m or 3m +1.