Thank you very much for the answer stated below:
A perfect square will have an even number of repetitions of each prime factor.
180 = 2*2*3*3*5
e have 2^2 and 3^2, but we need another 5. So, 180*5 = 900 = 30^2 = (2*3*5)^2
You're welcome! I'm glad the explanation was helpful. In the given example, we want to determine if 180 is a perfect square.
To do this, we need to analyze the prime factorization of 180, which is 2^2 * 3^2 * 5.
For a number to be a perfect square, it must have an even number of repetitions of each prime factor.
In the prime factorization of 180, we have 2^2 and 3^2, which means that we have even repetitions of these prime factors. However, we are missing another 5 to make the repetitions even. So, to make the prime factorization balanced, we need to multiply 180 by 5.
When we multiply 180 by 5, we get 900. Now, let's analyze the prime factorization of 900: 2^2 * 3^2 * 5^2.
We can see that 900 has even repetitions of all the prime factors.
So, 900 is a perfect square. In fact, it is equal to 30^2 or (2 * 3 * 5)^2.
Therefore, 180*5 = 900 is a perfect square.