Question

use euclid's division Lemma to show that cube of any positive integer is either of the form 9 q, 9q + 1 or 9q + 8 for some integer q

Answer 1

Let a be any positive integer and b = 3
a = 3q + r, where q ≥ 0 and 0 ≤ r < 3
  
Therefore, every number can be represented as these three forms. There are three cases.
Case 1: When a = 3q, 
  
Where m is an integer such that m =   
Case 2: When a = 3q + 1,
a 3 = (3q +1) 3 
a 3 = 27q 3 + 27q 2 + 9q + 1 
a 3 = 9(3q 3 + 3q 2 + q) + 1
a 3 = 9m + 1 
Where m is an integer such that m = (3q 3 + 3q 2 + q) 
Case 3: When a = 3q + 2,
a 3 = (3q +2) 3 
a 3 = 27q 3 + 54q 2 + 36q + 8 
a 3 = 9(3q 3 + 6q 2 + 4q) + 8
a 3 = 9m + 8
Where m is an integer such that m = (3q 3 + 6q 2 + 4q) 
Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.