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1008 divided by which digit number gives a perfect square?

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Last updated date: 01st Mar 2024
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IVSAT 2024
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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