Question

Find the smallest number by which 14700 must be divided so that it becomes a perfect square.

Answer 1

Hint: First we will start dividing the given number by \[2,3,4,5,...\] one at a time. Then we will check the quotient that we got from the division, whether it is a perfect square or not. A perfect square is nothing but when we multiply any whole number by the same number, we will get a whole number.

Complete step-by-step solution:
The given number is \[14700\], we aim to find the smallest number which will be a perfect square number on dividing the given number.
Now let’s start dividing \[14700\]. We will start with two since we need to find the smallest number.
\[14700 \div 2 = 7350\]
We got a quotient \[7350\]. Now let us check whether it is a perfect square number or not.
We know that a whole number is called a perfect square if we get a whole number on squaring it.
Now let us find the square root of \[7350\].
\[\sqrt {7350} \simeq 85.73\]
We got a decimal number, thus \[2\] cannot be the answer.
Let us divide \[14700\] by three.
\[14700 \div 3 = 4900\]
We got a quotient\[4900\]. Now let us check whether it is a perfect square number or not.
We already know that a whole number is called a perfect square if we get a whole number on squaring it.
Now we are going to find the square root of\[4900\].
\[\sqrt {4900} = 70\]
That is \[{70^2} = 70 \times 70 = 4900\].
Therefore, \[4900\] is a perfect square number.
Thus, the smallest number on dividing \[14700\] gives a perfect square number is \[3\].

Note: Since it is given that we need to find the smallest number, we started with two. Any number divided by one will give us the same number. Here, \[14700\] is not a perfect square number hence one cannot be the answer. The next smallest number that gives the perfect square is two. Here we could also use the prime factorisation for finding the perfect square number.