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:

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Step-by-step explanation:

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Answer 2

Answer:

Step-by-step explanation:

Let a be any positive integer and b = 3

Using Euclid's division lemma -:

a = 3q + r, where q ≥ 0 and 0 ≤ r < 3

 

Therefore, every number can be represented as these three forms. There are three cases.

Case 1: When a = 3q, 

 Where m is an integer such that m =  (3q)³ = 27q³

9(3q³) = 9m

Case 2: When a = 3q + 1,

a³ = (3q +1)³ 

a³ = 27q³ + 27q² + 9q + 1 

a³ = 9(3q³ + 3q² + q) + 1

a³ = 9m + 1 

Where m is an integer such that m = (3q³ + 3q²+ q) 

Case 3: When a = 3q + 2,

a³ = (3q +2)³ 

a³ = 27q³ + 54q² + 36q + 8 

a³ = 9(3q³ + 6q² + 4q) + 8

a³ = 9m + 8

Where m is an integer such that m = (3q² + 6q²+ 4q) 

Therefore-;

The cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.