use Euclid's division lemma to show that the cube of any positive integer is of the form 9m,9m+1,9m+8
Answers
Refer to the attachment........
Cube of any positive integer is of the form 9m,9m+1,9m+8
Step-by-step explanation:
as per Euclid's division lemma
any integer can be represented in the form
of a = bq + r where r < b
let say b = 3 then r = 0 , 1 , 2
a = 3q , 3q + 1 , 3q + 2
a³ = (3q)³ = 27q³ = 9 ( 3q³ ) = 9 m
a³ = (3q + 1 )³ = 27q³ + 1 + 27q² + 9q = 9(3q³ + 3q² + q) + 1
= 9m + 1
a³ = (3q + 2 )³ = 27q³ + 8 + 54q² + 36q = 9(3q³ + 18q² + 12q) + 8
= 9m + 8
Hence cube of any positive integer is of the form
9m , 9m+1 or 9m + 8
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