Question

Use division algorithm to show that the cube of any positive integer is of the form 9 m,
9m + 1 or 9m + 8​

Answer 1

Step-by-step explanation:

this is your question solution

Answer 2

Let a be any positive integer where b=3

So, a=bq+r=3q+r

Where r is equal to or greater than zero but less than the value of b

So all possible values of r : 0,1,2

If r=0

a=3q

Cube on both sides

a^3=(3q)^3

a^3=27q^3

a^3=9(3q^3)

So, a^3=9m (3q^3=m)

If r=1

a=3q+1

Cube on both sides

a^3=(3q+1)^3

a^3=(3q)^3 + (1)^3 +3(3q)^2(1)+3(3q)(1)^2

a^3=27q^3+1+27q^2+9q

a^3=27q^3+27q^2+9q+1

a^3=9(3q^3+3q^2+q)+1

So, a^3=9m+1 (3q^3+3q^2+q = m)

If r=2

a=3q+2

Cube on both sides

a^3=(3q+2)^3

a^3=(3q)^3 + (2)^3 +3(3q)^2(2)+3(3q)(2)^2

a^3=27q^3+8+54q^2+36q

a^3=27q^3+54q^2+36q+8

a^3=9(3q^3+6q^2+4q)+8

So,a^3=9m+8 (3q^3+6q^2+4q = m)

Hope it will help u