Question

The smallest number by which $396$ must be multiplied so that the product becomes a perfect square is:

Answer 1

To find the smallest number to multiply and to find the number this will form a perfect square. For that we will first factorize the number given in the question by dividing the number given till the remainder is zero and then we will take the divisors and check if each of them has been multiplied three times or not and the number which is missing any one of the three numbers is the smallest number to be multiplied.

Complete step by step solution:
The factorization we are going to use is prime factorization; prime factorization is a process in which the numbers are prime numbers in multiple of three as given below:
$396={{x}^{2}}{{y}^{2}}{{z}^{2}}..$
where $x,y,..$ are prime numbers that are formed by dividing the number in terms of prime numbers.
So let us divide the number in smallest numbers as:

& 02\left| 396 \right. \\\ & 02\left| 196 \right. \\\ & 03\left| 99 \right. \\\ & 03\left| 33 \right. \\\ & 11\left| 11 \right. \\\ & 1 \\\ \end{aligned}$$ Therefore, the prime factorization of the number given in the question is calculated as: $$\Rightarrow 396=2\times 2\times 3\times 3\times 11$$ Hence, by the above sequence we can see that the prime number 2 has two multiple making it a square of 2 similarly 3 also has a square multiple but 11 only has only multiple making it not a perfect square therefore, for the number 11 to be a perfect square we need to multiply the rest of the numbers with one more 11 thus making the final factorization product as $$2\times 2\times 3\times 3\times 11$$. Now placing the numbers in the formula of $$396={{x}^{2}}{{y}^{2}}{{z}^{2}}..$$ where $$x,y$$ and $$z$$ are taken as $$2,3$$ and $$11$$ forming the multiple of numbers as: $$\Rightarrow 396={{x}^{2}}{{y}^{2}}{{z}^{2}}$$ $$\Rightarrow 396=2\times 2\times 3\times 3\times 11$$ Now multiplying an extra 11, we get the new value as: $$\Rightarrow 4356={{2}^{2}}{{3}^{2}}{{11}^{2}}$$ **Therefore, the lowest number multiplied to the factors of the numbers to form a perfect square is $$11$$.** **Note:** In prime number factorization, we use the prime numbers, prime numbers are numbers which are either divided by themselves or by one. It can also be said one number LCM where you need to find factors in multiple of two.