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**Hint:**In this type of question we have to use the concept of factorisation. Prime factorisation is the one of the methods of finding prime factors for the given number. We start by dividing the number by the first prime number that is 2 if you can divide evenly. Continue dividing by 2 until you cannot divide evenly anymore. Now, try dividing by the next prime number i.e. 3. The goal is to get a quotient of 1.

**Complete step by step answer:**

Now, we have to find the prime factorisation of the number 13500.

We know that, the prime factorisation of the number starts by dividing the number by the first prime number i. e. 2

Let us start by dividing 13500 by 2

\[\begin{align}

& \Rightarrow 13500\div 2=6750 \\

& \Rightarrow 6750\div 2=3375 \\

\end{align}\]

Now, we know that 3375 is an odd number and hence not divisible by 2. So let us try for the next prime number that is 3.

\[\begin{align}

& \Rightarrow 3375\div 3=1125 \\

& \Rightarrow 1125\div 3=375 \\

& \Rightarrow 375\div 3=125 \\

\end{align}\]

We can observe that the number 125 is not divisible by 3 so we try for the next prime number that is 5.

\[\begin{align}

& \Rightarrow 125\div 5=25 \\

& \Rightarrow 25\div 5=5 \\

& \Rightarrow 5\div 5=1 \\

\end{align}\]

So here, we get the quotient is equal to 1 and hence we have to stop here.

We can express this prime factorisation as follows:

\[\begin{align}

& 2\left| \!{\underline {\,

13500 \,}} \right. \\

& 2\left| \!{\underline {\,

6750 \,}} \right. \\

& 3\left| \!{\underline {\,

3375 \,}} \right. \\

& 3\left| \!{\underline {\,

1125 \,}} \right. \\

& 3\left| \!{\underline {\,

375 \,}} \right. \\

& 5\left| \!{\underline {\,

125 \,}} \right. \\

& 5\left| \!{\underline {\,

25 \,}} \right. \\

& 5\left| \!{\underline {\,

5 \,}} \right. \\

& \left| \!{\underline {\,

1 \,}} \right. \\

\end{align}\]

Therefore, prime factorisation of \[13500\] is \[2\times 2\times 3\times 3\times 3\times 5\times 5\times 5\].

**Note:**In this type of question students may make mistakes in divisibility by taking a bigger prime number first or by starting with the number which is totally wrong and hence leads to a wrong answer. Students have to remember that in prime factorisation we have to start with the first prime number i.e. 2. Also to check divisibility (that is whether the number is divisible by 2, 3, 5 and so on or not) students may use the rules of divisibility.

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