Question

The cube of a positive integer
is always of the form 9m+k where
m is a whole number. What are
the possible value of k​

Answer 1

Given:

The cube of a positive integer  is always of the form 9m+k where

m is a whole number

To Find:

The possible values of k.

Solution:

We can represent any number in any form as we like.

We can represent it in the form of a multiple of 3.

Let the number be P.

• P  = 3q + r , q is the quotient when P is divided by 3 and r is the remainder. r ∈ { 0, 1, 2}

Now Let take Cube of P :

• P³ = (3q + r)³ = (3q)³ + r³ + 3 (3q)²r + 3(3q)r²
• P³ = 27q³ + 27q²r + 9qr² + r³

We can easily observe that 27q² + 27q²r + 9qr² = 9 x ( a number ) ,

since it is divisble by 9.

Therefore we can rewrite P as :

• P³ = 9m + r³

Comparing with the expression given in question,

• k = r³

• r³ ∈ { 0³, 1³, 2³} = {0, 1, 8}
• k can take the values 0 , 1 and 8.

The possible values of k are 0, 1 and 8.

Answer 2

Answer:The cube of a positive integer

is always of the form 9m+k where

m is a whole number. What are

the possible value of k​

Step-by-step explanation: hdfn

5+602+85