Question

Find the square root of \[225\].
A. $12$
B. $13$
C. $15$
D. None of these

Answer 1

Hint: First, write the prime factorization of the given number. Then, we have to pair the prime factors. Also, we have to take the product of prime factors, and then we choose only one prime factor out of every pair. Finally we get the required answer.

Complete step-by-step solution:
Given number is \[225\].
Find the smallest prime number which divides \[225\].
Since, $3$ divides \[225\] as the sum of digits of \[225\] is $9$, which is divisible by $3$. So, we can divide \[225\] by $3$.
Now, we have to divide \[225\] by $3$ and write the quotient below \[225\].
\[\begin{array}{*{20}{c}}
  {3\left| \!{\underline {\,
  {225} \,}} \right. } \\
  {{\text{ }}75}
\end{array}\]
Now, we have to find the smallest prime number which divides $75$.
Since, $3$ divides $75$ as the sum of the digits of $75$ is $12$, which is divisible by $3$. So, we can divide $75$ by $3$.
As, we Divide $75$ by $3$ and write the quotient below $75$.
\[\begin{array}{*{20}{c}}
  {3\left| \!{\underline {\,
  {225} \,}} \right. } \\
  {3\left| \!{\underline {\,
  {75} \,}} \right. } \\
  {{\text{ }}25}
\end{array}\]
Now, we have to find the smallest prime number which divides $25$.
Since, $5$ divides $25$ as last digit is $5$. So, we can divide $25$ by $5$.
Divide $25$ by $5$ and write the quotient below $25$.
\[\begin{array}{*{20}{c}}
  {3\left| \!{\underline {\,
  {225} \,}} \right. } \\
  {3\left| \!{\underline {\,
  {75} \,}} \right. } \\
  {\begin{array}{*{20}{c}}
  {5\left| \!{\underline {\,
  {25} \,}} \right. } \\
  {{\text{ }}5}
\end{array}}
\end{array}\]
So, the prime factorisation of \[225\] is
$225 = 3 \times 3 \times 5 \times 5$
Pair the prime factors.
$225 = \underline {3 \times 3} \times \underline {5 \times 5} $
Now, take the product of prime factors, choosing one factor out of every pair.
Here we take $3$ and $5$ out of pairs and multiply them.
So, $\sqrt {225} = 3 \times 5 = 15$
Therefore, the square root of $225$ is $15$.

Hence, option (C) is correct.

Note: Points to be remember:
Steps for finding the square root of a number:
Step 1: Write the prime factorization of the number.
Step 2: Pair the prime factors.
Step 3: The product of prime factors, we can choose one factor out of every pair.
Divisibility rule of $3$:
If the number is divisible by $3$ then, the sum of all digits of a number is divisible by $3$.
Divisibility rule of $5$:
If the last digit of the number is either $0$ or $5$, then the number is divisible by $5$.