Question

By which smallest number $ 3388 $ should be multiplied to make it a perfect square? Also find the square root of the perfect square.

Answer 1

Hint: To find the required smallest number, we will use the prime factorization method. We will write the given number $ 3388 $ as the multiple of primes. After that it will be written in the form of a group of two primes if possible.

Complete step-by-step answer:
To solve the given problem, we must know the prime factorization method. By using the method of prime factorization, we can express the given number as a product of prime numbers. Therefore, we will write the given number $ 3388 $ as the product of primes. Let us do the prime factorization of $ 3388 $ . Note that $ 3388 $ is an even number so we can start prime factorization with prime number $ 2 $ .

$ 2 $	$ 3388 $
$ 2 $	$ 1694 $
$ 7 $	$ 847 $
$ 11 $	$ 121 $
$ 11 $	$ 11 $
	$ 1 $

Therefore, we can write $ 3388 = 2 \times 2 \times 7 \times 11 \times 11 $ . Here we are dealing with a perfect square so we have to write the obtained factorization in the form of a group of two primes if possible. So, we can write $ 3388 = \left( {2 \times 2} \right) \times 7 \times \left( {11 \times 11} \right) \cdots \cdots \left( 1 \right) $ . To find a perfect square root, we have to take one number from each group of two but here we can see that a single $ 7 $ cannot be written as a group of two.
Let us multiply by $ 7 $ on both sides of the equation $ \left( 1 \right) $ . So, we can write
 $ 3388 \times 7 = \left[ {\left( {2 \times 2} \right) \times 7 \times \left( {11 \times 11} \right)} \right] \times 7 $
 $ \Rightarrow 3388 \times 7 = \left( {2 \times 2} \right) \times \left( {7 \times 7} \right) \times \left( {11 \times 11} \right) $
Here we can take one number from each group of two. So we will get the perfect square root. Hence, the required smallest number is $ 7 $ . Hence, $ 3388 $ should be multiplied by $ 7 $ to be a perfect square number.
Let us take one number from each group of two to find the square root. So, we can write
 $ \sqrt {3388 \times 7} = 2 \times 7 \times 11 = 154 $
Hence, $ 154 $ is square root of the perfect square number.

Note: Remember that if the number is even then it is divisible by $ 2 $ . Double the last digit of the number and subtract the doubled number from the remaining number (remaining digits). If the result is divisible by $ 7 $ then that number is divisible by $ 7 $ . Note that here we will consider positive differences. In the given problem, $ 847 $ is divisible by $ 7 $ because double of last digit $ 7 $ is $ 14 $ and positive difference of remaining number (remaining digit) $ 84 $ and $ 14 $ is $ 70 $ and the number $ 70 $ is divisible by $ 7 $ .