Question

Use Euclid's division lemma to show that the cube of any positive integer is of the form
9 m, 9m + 1 or 9m +8.
A:
1.1.
1​

Answer 1

Answer:

using Euclid division algorithm, we know that a = bq + r, Osrsb—- (1)

Let a be any positive integer, and b = 3.

Substitute b = 3 in equation (1)

a = 3q + r where o srs3, r= 0, 1, 2

If r= 0, a = 39

Cube the value, we get 23 = 2703 23 = 9(393), where m=393 —(2)

If r = 1, a = 3q+1 Cube the value, we get 23 = (39+13 a3 = (2773 + 2772 +9q+1) 23 = 9/393 +382 + 1) +1, where m= 393 +372 +9—(3)

If r= 2, a = 39+2

Cube the value, we get an2 = 139+213 43 = (27q2 + 5302 + 36q+8) 23 = 9393 +692 + 49) + 8, where m=393 +692 +49—-(4)

From equation 2, 3 and 4,

The cube of any positive integer is of the form 9m, 9m + 1 or 9m+8.