Answer

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**Hint:**We have to solve this type of question with the concept of factorisation. We have to find the factors of 1008. Write it in a single expression and look for the single digit number which is not paired. Thus, that number is the required answer.

**Complete step by step solution:**

According to the question, we are asked to find the single digit number which on dividing with 1008 gives a perfect square.

We have been given the number 1008.

We have to first find the factors of 1008.

Let us find the factors of 1008.

Factoring is a method where we can find the factor of a number by dividing the given number by divisible prime number.

By the factoring method, we get

\[\begin{align}

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

1008 \,}} \right. \\

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

504 \,}} \right. \\

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

252 \,}} \right. \\

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

126 \,}} \right. \\

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

63 \,}} \right. \\

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

21 \,}} \right. \\

& 7\left| \!{\underline {\,

7 \,}} \right. \\

& 1\left| \!{\underline {\,

1 \,}} \right. \\

\end{align}\]

Now, we get the factors of 1008 which are 2, 2, 2, 2, 3, 3 and 7.

So we can write 1008 as

\[1008=2\times 2\times 2\times 2\times 3\times 3\times 7\]

Here, we have to look for the pair of terms.

There are two pairs of 2 and one pair of 3.

But we observe that the number 7 is not having a pair.

Hence, on dividing the given number by 7, we should get a perfect square.

We know that \[144\times 7\] is equal to 1008.

\[\Rightarrow \dfrac{1008}{7}=\dfrac{144\times 7}{7}\]

On cancelling 7 from the numerator and denominator of the right-hand side of the equation, we get

\[\dfrac{1008}{7}=144\]

But we know that \[144={{12}^{2}}\].

On taking the square root of 144, we get

\[\sqrt{144}=\sqrt{{{12}^{2}}}\]

Using the property \[\sqrt{{{x}^{2}}}=x\], we get

\[\sqrt{144}=12\]

12 is the perfect square root of 144.

Hence, on dividing the given number by 7, we get a perfect square.

**Therefore, 1008 divided by the single digit number 7 gives a perfect square.**

**Note:**We can also solve this question by trial and error method. Take the entire digit from 1 to 9 and divide with the given number. Check for which number on dividing with 1008, we get a perfect square.

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