Question 5

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

Answer

Let a be any positive integer and b = 3. Then, by euclid’s algorithm

a = 9q + r, where q ≥ 0 and r = 0, 1, 2 [ 0 ≤ r ≤ b ]

For r = 0, x = 3q, or

For r = 1, x = 3q +1

For r = 2, x = 3q + 2

Now by taking the cube of all the three above terms, we get,

Case (i) : when r = 0, then,

x³ = (3q)³ = 27q³ = 9 (3q³) = 9m; where m = 3q

Case (ii) : when r = 1, then,

x³ = (3q +1)³ = (3q)³ + 1³ +3 × 3q × 1 (3q + 2 ) = 27q³ + 1 +27q² + 9q

Taking 9 as common factor, we get,

x³ = 9 (3q³ +3q² + q) +1

Putting (3q³ + 3q² + q) = m, we get,

x³ = 9m + 1

Case (iii) : when r = 2, then,

x³ = (3q + 2)³ = (3q)³ + 2³ + 3 × 3q × 2 (3q + 2) = 27q³ + 54q² + 36q + 8

Taking 9 as common factor , we get

x = 9 (3q + 6q + 4q) + 8

Putting (3q + 6q + 4q) = m, we get,

x = 9m + 8,

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

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