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

By Ritesh|Updated : November 10th, 2022

Using Euclid's division lemma it is proved that the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8. Steps to prove that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8:

Step 1: Show that any positive integer's cube has the shape 9m.

According to Euclid's division lemma, if two positive integers a and b exist, then there must also exist two distinct integers q and r such that

a = bq + r, 0 ≤ r < b

Now suppose b = 3, then 0 ≤ r < 3

So, possible values of r = 0, 1, 2

According to Euclid's division lemma, the equation for r = 0 is;

a = 3q

When we cube all sides, we have;

a3 = (3q)3

a3 = 27q3

a3 = 9 (3q)3

a3 = 9m, where m = 3q3

Step 2: Show that any positive integer's cube has the form 9m + 1.

The equation given by Euclid's division lemma is; for, r = 1.

a = 3q + 1

On cubing both sides, we get;

a3 = (3q + 1)3

a3 = 27q3 + 1 + 27q2 + 9q [(a + b)3 = a3 + b3 + 3a2b + 3ab2]

a3 = 27q3 + 27q2 + 9q + 1

a3 = 9 (3q3 + q2 + q) + 1

a3 = 9m + 1 where m = 3q3 + q2 + q

Step 3: Show that a cube of any positive integer has the form 9m + 8 for r = 2, which results in the equation;

a = 3q + 2

On cubing both sides, we get;

a3 = (3q + 2)3

a3 = 27q3 + 8 + 54q2 + 36q [(a + b)3 = a3 + b3 + 3a2b + 3ab2]

a3 = 27q3 + 54q2 + 36q + 8

a3 = 9 (3q3 + 6q2 + 4q) + 8

a3 = 9m + 8 where m = 3q3 + 6q2 + 4q

Hence, it is proved that the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8

Summary:

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

Using Euclid's division lemma it is proved that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

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