Question

Use Euclid's Division Algorithm to show that the cube of any positive integer is either of the 9m, 9m + 1 or 9m + 8 for some integer m

Answer 1

Let a and b be two positive integers such that a is greater than b; then:

a = bq + r; where q and r are positive integers and 0 ≤ r < b.

Taking b = 3, we get:

a = 3q + r; where 0 ≤ r < 3

⇒ Different values of integer a are 3q, 3q + 1 or 3q + 2.

Cube of 3q = (3q)³ = 27q³ = 9(3q³) = 9m; where m is some integer.

Cube of 3q + 1 = (3q + 1)³

= (3q)³ + 3(3q)² x 1 + 3(3q) x 1² + 1³

[Q (q + b)³ = a³ + 3a²b + 3ab² + 1]

= 27q³ + 27q² + 9q + 1

= 9(3q³ + 3q² + q) + 1

= 9m + 1; where m is some integer.

Cube of 3q + 2 = (3q + 2)³

= (3q)³ + 3(3q)² x 2 + 3 x 3q x 2² + 2³

= 27q³ + 54q² + 36q + 8

= 9(3q³ + 6q² + 4q) + 8

= 9m + 8; where m is some integer.

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

Hence the required result.