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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 BYJU'S Exam Prep
Updated on: September 25th, 2023
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
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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.