Q. 54.1( 556 Votes )

Use Euclid’

Answer :

Let a be any positive integer. Then, it is of the form 3q or, 3q + 1 or, 3q + 2.

We know that according to Euclid's division lemma:
a = bq + r So, we have the following cases:

Case I When a = 3q

In this case, we have

a3 = (3q)3 = 27q3 = 9(3q3 ) = 9m, where m = 3q3

Case II When a = 3q + 1

In this case, we have

a3 = (3q + 1)3

27q3 + 27q2 + 9q + 1

9q(3q2 + 3q + 1) + 1

a3 = 9m + 1, where m = q(3q2 + 3q + 1)

Case III When a = 3q + 2

In this case, we have

a3 = (3q + 1)3

27q3 + 54q2 + 36q + 8

9q(3q2 + 6q + 4) + 8

a3 = 9m + 8, where m = q(3q2 + 6q + 4)

Hence, a3 is the form of 9m or, 9m + 1 or, 9m + 8

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses
RELATED QUESTIONS :

Every odd integerRD Sharma - Mathematics

Every even integeRD Sharma - Mathematics

The LCM of two nuRS Aggarwal - Mathematics

A number when divRS Aggarwal - Mathematics

If n is an odd inNCERT - Maths Exemplar

Show that the squMathematics - Board Papers

The remainder wheRD Sharma - Mathematics