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.

Answer 1

Let ‘a’ and ‘b’ are positive integers, 

a = b × q + r 

Let b = 3. 

∴ a = 3q + r 

(i) If r = 0 then a = 3q 

(ii) If r = l then a = 3q + 1 

(iii) If r = 2 then a = 3q + 2 

If we consider cubes of these, 

(i) If a = 3q, then 

a³ = (3)³ = 27q³ = 9(3q)³ = 9m (∵ m = 3q²) 

(ii) If a = 3q + 1, then (a)³ = (3q + 1)³ 

= 27q³ + 1 + 27q³ + 9q 

= 9(3q³ + 3q² + q) + 1 

= 9m + 1 

(∵ m = 3q³+ 3q² + q) 

iii) If a = 3q + 2, then 

(a)³ = (3q + 2)³ = 27q³ + 54q² + 36q + 8 

= 9(3q³ + 6q² + 4q) + 8 

= 9m + 8 

∴ Cube of any positive integer is of the form 9m, 9m + 1, 9m + 8.