Question

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 1

Answer:

a = 9m + 1 [ Where m = 3q³ + 3q² + q ) . Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8. Hence, it is proved .27

Answer 2

Let assume a and b where a be any positive number and b is equal to 3.

As per Euclid’s Division Lemma

• a = bq + r (0 ≤ r < b)
• a = 3q + r (0 ≤ r < 3)

Hence r → 0, 1 or 2.

Case 1: When r = 0, the equation becomes

• a = 3q
• a³ = (3q)³
• a³ = 27 q³
• a³ = 9 (3q³)
• a³ = 9m where m = 3q³

Case 2: When r = 1, the equation becomes

• a = 3q + 1
• a³ = (3q + 1)³
• a³ = (3q)³ + 1³ + 3 × 3q × 1(3q + 1)
• a³ = 27q³ + 1 + 9q × (3q + 1)
• a³ = 27q³ + 1 + 27q² + 9q
• a³ = 27q³ + 27q² + 9q + 1
• a³ = 9 ( 3q³ + 3q² + q) + 1
• a³ = 9m + 1 Where m = ( 3q³ + 3q² + q)

Case 3: When r = 2, the equation becomes

• a = 3q + 2
• a³ = (3q + 2)³
• a³ = (3q)³ + 2³ + 3 × 3q × 2 (3q + 1)
• a³ = 27q³ + 8 + 54q² + 36q
• a³ = 27q³ + 54q² + 36q + 8
• a³ = 9 (3q³ + 6q² + 4q) + 8
• a³ = 9m + 8 Where m = (3q³ + 6q² + 4q)

Therefore a can be any of the form 9m or 9m + 1 or, 9m + 8.