NCERT Solutions for Class 10 Maths | Chapter: Real Numbers | Concept: Applicatoin of Euclid's Division Algorithm
Question 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.]
Explanatory Answer
Let ‘x’ be any positive integer.
Let us apply Euclid’s division algorithm on ‘x’ with 3 as the divisor.
x = 3q + r , 0 ≤ r < 3
∴ x = 3q + 0 or 3q + 1 or 3q + 2
If x = 3q, x2 = 9q2 = 3(3q2)
Let 3q2 = m
So, x2 = 3m -------------(1)
If x = 3q + 1, x2 = (3q + 1)2 = 9q2 + 6q + 1
x2 = 3(3q2 + 2q) + 1
Let 3q2 + 2q be ‘m’
So, x2 = 3m + 1 --------(2)
If x = 3q + 2, x2 = (3q + 2)2 = 9q2 + 12q + 4
x2 = 9q2 + 12q + 3 + 1
= 3(3q2 + 4q + 1) + 1
Let 3q2 + 4q + 1 be m
x2 = 3m + 1 --------------- (3)
From (1), (2), and (3) we can conclude that the square of a positive number, x2 is of the form 3m or 3m + 1