Question

The number ${3^{13}} - {3^{10}}$ is divisible by
(i)2 and 3 only
(ii)3 and 10 only
(iii)2,3,and 10 only
(iv)all 2,3 and 13

Answer 1

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