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**Hint:**We need to reduce the expression to a simpler form by taking out the common element ${3^{10}}$and then calculate the value of $({3^3} - 1)$ and by checking the divisibility of the obtained expression we get our answer

**Complete step by step answer:**

The given expression is ${3^{13}} - {3^{10}}$

In order to reduce it into its simpler form,lets take ${3^{10}}$ as common

$\begin{gathered}

\Rightarrow {3^{10}}({3^3} - 1) \\

\\

\end{gathered} $

Step 2:Now let’s make this expression much simpler.

We know that ${3^3} = 27$

Hence we get,

$\begin{gathered}

\Rightarrow {3^{10}}(27 - 1) \\

\Rightarrow {3^{10}}(26) \\

\end{gathered} $

Step 3:From this it is clearly understood that the expression is divisible by 3.

The expression can also be written as ${3^{10}}(2*13)$

This shows that it is a multiple of 2 and 13.

Hence it is divisible by 2 and 13 also.

Therefore the given expression is divisible by 2,3 and 13.

**The answer is option d.**

**Note:**1)In problems of these kinds, it is enough if we reduce the expression to its simplest form. There is no need to keep calculating the values of ${3^{13}}$ and ${3^{10}}$. This will take a lot of time and there are chances of making mistakes.

2)You need to read the question more carefully as many students write ${3^{13}} - {3^{10}} = {3^3}$, which is completely wrong.

3)Many students get distracted by the choices provided and end up choosing that it is divisible by 10 as the expression consists of the term

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