Exercise 3.1, Q3,4 and 5

Exercise 3.1 – Q3 Solution

Question

Write the cubes of all natural numbers between 1 and 10 and verify the following statements:

1. Cubes of all odd natural numbers are odd.
2. Cubes of all even natural numbers are even.

Solution

Verification

(i) Cubes of all odd natural numbers are odd:

Odd numbers: \(1,3,5,7,9\)

Their cubes are: \(1,27,125,343,729\)

All these numbers are odd.

Hence, the cubes of all odd natural numbers are odd.

(ii) Cubes of all even natural numbers are even:

Even numbers: \(2,4,6,8,10\)

Their cubes are: \(8,64,216,512,1000\)

All these numbers are even.

Hence, the cubes of all even natural numbers are even.

Exercise 3.1 – Q4 Solution

Question

Write the cubes of 5 natural numbers which are multiples of 3 and verify the following statement:

“The cube of a natural number of the form \(3n+1\) is a natural number of the same form.”

Solution

Five multiples of 3: \(3,\;6,\;9,\;12,\;15\)

Verification of the Statement

Take five natural numbers of the form \(3n+1\):

\(1,\;4,\;7,\;10,\;13\)

Hence, the cube of a natural number of the form \(3n+1\) is also of the form \(3n+1\). Therefore, the statement is verified.

Exercise 3.1 – Q5 Solution

Question

Write the cubes of 5 natural numbers which are multiples of 7 and verify the following statement:

“The cube of a multiple of 7 is a multiple of \(7^3\).”

Solution

Five multiples of 7: \(7,\;14,\;21,\;28,\;35\)

Since every cube is exactly divisible by \(343=7^3\), the cube of every multiple of 7 is a multiple of \(7^3\).

Hence, the statement is verified.