Question

Use Euclid's division lemma, to show that the cube of any positive integer is of the form 3p or 3p +1 or 3p +2 for any integer 'p'.​

Answer 1

༒ Qυєѕтiσи ༒

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Use Euclid's division lemma, to show that the cube of any positive integer is of the form 3p or 3p +1 or 3p +2 for any integer 'p'.

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★彡 Sσℓυтiσи 彡★

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Let 'a' be positive integer

a = bq + r, 0 < r < b

b = 3 so r = 0, 1, 2

Then 'a' can be of the forms

3q + 0, 3q + 1, 3q + 2

Case ( 1 ) :-

When a = 3q

a³ = (3q)³ = 3 (9q³)

= 3p where p = 9q³

Case ( 2 ) :-

When a = 3q +1

a³ = (3q + 1)³

a³ = (3q)³ + 3 (3q) (1) (3q+1) + (1)³

a³ = 3 [ 9q³ + 3q (3q + 1) ] + 1

a³ = 3p + 1

Where p = 9q³ + 3q (3q + 1)

Case ( 3 ) :-

When a = 3q + 2

a³ = (3q + 2)³

a³ = (3q)³ + 3(3q) (2) (3q + 2) + (2)³

a³ = 3 [ 9q³ + 6q (3q + 2) ] + 2

a³ = 3p + 2

Where p = 9q³ + 6q(3q + 2)

So the cube of any positive integer is of the form 3p or 3p + 1 or 3p + 2 for any integer 'p'.

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