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

Question

Use Euclid&#x2019;s division lemma to show that the cube of any positive integer is of the form 9 m , 9 m + 1 or 9 m + 8.

Philoid · Accepted Answer

By taking,’ a’ as any positive integer and b = 3.

Applying Euclid’s algorithm

a = 3q+r

Here,

So, total possible forms are

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

when a = 3q,

a³ = (3q)³ = 27q³ = 9(3q³) = 9m,

where m is an integer such that m = 3q³

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)

when a = 3q+2,

a³ = (3q+2)³

a³ = 27q³+54q²+36q+8

a³ = 9(3q³+6q²+4q)+8

a³ = 9m+8

Hence,

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