Number of Perfect Square Factors of 4500

To find the number of perfect square factors of a given number, we need to analyze the prime factorization of that number. In this case, we will analyze the prime factorization of 4500.

Prime Factorization of 4500

To find the prime factorization of 4500, we can start by dividing it by the smallest prime number, which is 2.

4500 ÷ 2 = 2250

We continue dividing the result by 2 until we can no longer divide evenly.

2250 ÷ 2 = 1125
1125 ÷ 2 = 562.5 (Not divisible by 2)

Next, we move on to the next prime number, which is 3.

562.5 ÷ 3 = 187.5 (Not divisible by 3)

Then, we move on to the next prime number, which is 5.

187.5 ÷ 5 = 37.5 (Not divisible by 5)

Since 37.5 is not divisible by any prime numbers, we can conclude that the prime factorization of 4500 is:

4500 = 2^2 × 3^2 × 5^3

Perfect Square Factors

To determine the perfect square factors, we need to consider the exponents of the prime factors in the prime factorization.

In this case, the exponents of 2, 3, and 5 are 2, 2, and 3, respectively.

We can find the number of perfect square factors by adding 1 to each exponent and multiplying them together.

(2 + 1) × (2 + 1) × (3 + 1) = 3 × 3 × 4 = 36

Therefore, the number of perfect square factors of 4500 is 36.

Summary

- The prime factorization of 4500 is 2^2 × 3^2 × 5^3.
- To find the number of perfect square factors, we add 1 to each exponent and multiply them together.
- In this case, the number of perfect square factors is 36.