Use Euclid’s division lemma to show that the cube of any positive in

1.	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» 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.

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»

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.