NCERT Solutions for Class 10 Maths | Chapter: Real Numbers | Concept: Euclid's Division Algorithm Applications
Question: Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
Explanatory Answer
Let ‘a’ be any positive integer
Let us apply Euclid’s division lemma on ‘a’ with 3 as the divisor.
So, a = 3q + r , 0 ≤ r < 3
∴ a = 3q + 0 or 3q + 1 or 3q + 2
If a = 3q, a3 = (3q)3 = 27 q3 = 9(3q3)
Let 3q3 be m.
So, a3 = 9m ----------(1)
If a = 3q + 1, a3 = (3q + 1)3 = 27q3 + 27q2 + 9q + 1
a3 = 9(3q3 + 3q2 + q) + 1
Let 3q3 + 3q2 + q be m
So, a3 = 9m + 1 -----(2)
If a = 3q + 2, a3 = (3q + 2)3 = 27q3 + 54q2 + 36q + 8
a3 = 9(3q3 + 6q2 + 4q) + 8
Let 3q3 + 6q2 + 4q be m
So, a3 = 9m + 8 -----(3)
From (1), (2), and (3) we can conclude that the cube of a positive integer will be of the form 9m or 9m + 1 or 9m + 8