Question

State true or false: The cube of a two digit number maybe a three digit number.
(a) True
(b) False

Answer 1

Hint:
Here, we need to check whether the cube of a two digit number can be a three digit number or not. We will check the cube of the smallest two digit natural number and check if the result is a three digit number or not. If a number \[x\] is greater than a number \[y\], then the cube of \[x\] is also greater than \[y\]. Using this, we can prove whether the cubes of any two digit number can be a three digit number or not.

Complete step by step solution:
We will find the cube of the smallest two digit number.
We know that the smallest two digit number is 10.
Let us find the cube of 10.
The cube of a number \[x\] is given by \[x \times x \times x\]. It can be written using an exponent in the form \[x \times x \times x = {x^3}\], where \[x\] is the base and 3 is the exponent.
Therefore, we get the cube of 10 as
\[{10^3} = 10 \times 10 \times 10\]
Multiplying the terms in the expression, we get
\[ \Rightarrow {10^3} = 1000\]
Therefore, the cube of the smallest two digit number, that is 10, is 1000.
The number 1000 is a four digit number.
We know that if a number \[x\] is greater than a number \[y\], then the cube of \[x\] is also greater than \[y\].
Since 10 is the smallest two digit number, all the other two digit numbers are greater than 10.
Therefore, the cube of 10 is smaller than the cubes of the other two digit numbers.
Since the cube of 10 is a four digit number, the cubes of the other two digit numbers have at least four digits each.
Therefore, we can say that the cube of any two digit number is not a three digit number.

Thus, the given statement is false.
The correct option is option (b).

Note:
We used the term ‘natural number’ in the solution. A natural number is any integer that comes after 0. For example, 1, 2, 3, 4, 143, etc. are all natural numbers. The number 0 is not a natural number.