Question

By taking three different values of n verify the truth of the following statements:
(i) If n is even , then n³ is also even.
(ii) If n is odd, then n³ is also odd.
(iii) If n leaves remainder 1 when divided by 3, then n³ also leaves 1 as remainder when divided by 3.
(iv) If a natural number n is of the form 3p + 2 then n³ also a number of the same type.

Answer 1

(i)
Let the three even natural numbers be 2, 4 and 8.
Cubes of these numbers are:
$2^3=8,4^3=64,8^3=512$
By divisibility test, it is evident that $8,64\text{and}512$ are divisible by 2.
Thus, they are even.
This verifies the statement.

(ii)
Let the three odd natural numbers be 3, 9 and 27.
Cubes of these numbers are:
$3^3=27,9^3=729,{27}^3=19683$
By divisibility test, it is evident that $27,729\text{and}19683$ are divisible by 3.
Thus, they are odd.
This verifies the statement.

(iii)
Three natural numbers of the form (3n + 1) can be written by choosing $n=1,2,3...\text{etc.}$
Let three such numbers be $4,7\text{and}10.$
Cubes of the three chosen numbers are:
$4^3=64,7^3=343\text{and}{10}^3=1000$

Cubes of $4,7\text{and}10$ can expressed as:
$64=3\times 21+1$, which is of the form (3n + 1) for n = 21
$343=3\times 114+1$, which is of the form (3n + 1) for n = 114
$1000=3\times 333+1,$ which is of the form (3n + 1) for n = 333

Cubes of $4,7,\text{and}10$ can be expressed as the natural numbers of the form (3n + 1) for some natural number n. Hence, the statement is verified.

(iv)
Three natural numbers of the form (3p + 2) can be written by choosing $p=1,2,3...\text{etc.}$
Let three such numbers be $5,8\text{and}11.$
Cubes of the three chosen numbers are:
$5^3=125,8^3=512\text{and}{11}^3=1331$

Cubes of $5,8,\text{and}11$ can be expressed as:
$125=3\times 41+2$, which is of the form (3p + 2) for p = 41
$512=3\times 170+2$, which is of the form (3p + 2) for p = 170
$1331=3\times 443+2,$ which is of the form (3p + 2) for p = 443

Cubes of $5,8,\text{and}11$ could be expressed as the natural numbers of the form (3p + 2) for some natural number p. Hence, the statement is verified.