Hello, Amy!

I'm not familiar with Euclid's Division Lemma,

. . but I have a proof.

We want to show that the cube of any positive integer is: a multiple of 9,Show that the cube of any positive integer is of the form 9m, 9m+1 or 9m+8.

or one more than a multiple of 9, or eight more than a multiple of 9.

A positive integerNmust be in one of three forms:

. . . . 3a: . a multiple of three

. . 3a + 1: one more than a multiple of three

. . 3a + 2: two more than a multiple of three

(1) N = 3a

. . .Then .N³ .= .(3a)³ .= .27a³ .= .9(3a³)

. . .Hence, it is a multiple of 9.

(2) N = 3a + 1

. . .Then .N³ .= .(3a + 1)³ .= .27a³ + 27a² + 9a + 1 .= .9(3a³ + 3a² + a) + 1

. . .Hence, it is one more than a multiple of 9.

(3) N = 3a + 2

. . .Then .N³ .= .(3a + 2)³ .= .27a³ + 54a² + 36a + 8 .= .9(3a³ + 6a² + 4a) + 8

. . .Hence, it is eight more than a multiple of 9.

. . Q. E. D.