Question

Which $3$ digit number is both a square and a cube?

Answer 1

Hint:Here, in the given question, we need to find which $3$ digit number is both a square and a cube. To obtain the required answer, at first we will write a cube of all numbers from $1$ to $9$. We will write the cube till $9$ only, because the next digit is $10$ and the cube of $10$ is a four digit number. Side-by-side we will also write a square of numbers which are the same as a cube of a number. After this we will check which $3$ digit number is both a square and a cube.

Complete step by step answer:
As here we have to find a $3$ digit number, so we will write the cubes of numbers and side-by-side we will also write squares of numbers which are the same as a cube of a number.
${1^3} = 1 = {1^2} = 1$
As we can see, the square and cube of $1$ are the same, i.e., $1$. But it is not our correct answer because it is a one digit number.
${2^3} = 8$
$\Rightarrow {3^3} = 27$
$\Rightarrow {4^3} = 64 = {8^2} = 64$
As we can see, the square of $8$ is $64$ and the cube of $4$ is $64$. But it is not our correct answer because it is a two digit number.
${5^3} = 125$
$\Rightarrow {6^3} = 216$
$\Rightarrow {7^3} = 343$
$\Rightarrow {8^3} = 512$
$\therefore {9^3} = 729 = {36^2} = 729$
As we can see, the square of $36$ is $729$ and the cube of $9$ is $729$ and also, it is a three digit number, so it is our correct answer.

Note:To solve this type of question, one must know the square and cube of all the numbers. Remember that a square number is a number multiplied by itself and a cube number is a number multiplied by itself $3$ times. We can write a square number as $a \times a = {a^2}$ and a cube number as $a \times a \times a = {a^3}$.