Question

By what least number the given number be divided to get a perfect square number? In this case, find the number of the square is the new number 3380

Answer 1

Hint: The question can be solved by applying the concept of splitting the number as the product of its prime factors and then grouping each prime factor as an even power. If we find any prime factor not raised to even power, then we need to divide unless an even power is obtained. After these operations are performed, a square number is obtained. Now, to find the number whose square is obtained, we can just group them under one bracket and raise them to an even common power.

Complete step-by-step answer:
Now, let us begin the question by splitting the given number as the product of prime numbers as shown below,
\[ \Rightarrow 3380 = 2 \times 2 \times 5 \times 13 \times 13\]
\[ \Rightarrow 3380 = {2^2} \times 5 \times {13^2}\]
Now, clearly, 5 is not raised to an even power; thus, we need to divide the number by 5 to get,
\[ \Rightarrow 3380 = {2^2} \times {13^2}\]
Now, every prime factor of the number is raised to an even power. Thus, by dividing the number by 5 we get a square number.
Or 5 is the least number with which the number should be divided to obtain a perfect square number.
Now, we need to find the number x, whose square is the given number.
So, for that, we just group all the prime factors under the power two as shown below,
\[ \Rightarrow 3380 = {(2 \times 13)^2}\]
Now, after solving the bracket we get,
\[ \Rightarrow 3380 = {(26)^2}\]
\[ \Rightarrow x = (26)\]
Thus, the required number is 26.

Note: This question is based on the concept of splitting a number into its prime factors. One should be well versed with these concepts before solving this question. Do not commit calculation mistakes, and be sure of the final answer.