Courses
Courses for Kids
Free study material
Offline Centres
More
Store

1008 divided by which digit number gives a perfect square?

Last updated date: 10th Aug 2024
Total views: 390.3k
Views today: 6.90k
Verified
390.3k+ views
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.