The number 360 is divisible by 24 numbers.
What are Divisors?
Divisors, also known as factors, are numbers that divide another number exactly, leaving no remainder. For instance, the divisors of 10 are 1, 2, 5, and 10 because 10 can be divided evenly by each of these numbers.
Finding the Divisors of 360
There are two primary methods to determine the exact count of a number's divisors: by listing all of them or by using its prime factorization.
Method 1: Listing All Divisors
By listing all the numbers that divide 360 evenly, we can simply count them.
The Divisors of 360:
| Divisor | Divisor | Divisor | Divisor |
|---|---|---|---|
| 1 | 2 | 3 | 4 |
| 5 | 6 | 8 | 9 |
| 10 | 12 | 15 | 18 |
| 20 | 24 | 30 | 36 |
| 40 | 45 | 60 | 72 |
| 90 | 120 | 180 | 360 |
Counting these numbers reveals that 360 has 24 divisors.
Method 2: Using Prime Factorization
A more systematic and efficient way to find the number of divisors, especially for larger numbers, is by using its prime factorization. Prime factorization breaks a number down into its prime number building blocks.
Prime Factorization of 360:
The prime factorization of 360 is 2³ × 3² × 5¹. This means 360 can be expressed as 2 multiplied by itself three times, 3 multiplied by itself two times, and 5 multiplied by itself one time (since 5 is 5 to the power of 1).
Calculating the Number of Divisors:
To find the total number of divisors from the prime factorization, follow these steps:
- Identify the exponents of each prime factor. For 360 (2³ × 3² × 5¹), the exponents are 3, 2, and 1.
- Add 1 to each exponent:
- For the prime factor 2, the exponent is 3, so 3 + 1 = 4.
- For the prime factor 3, the exponent is 2, so 2 + 1 = 3.
- For the prime factor 5, the exponent is 1, so 1 + 1 = 2.
- Multiply the results together: 4 × 3 × 2 = 24.
This method confirms that 360 has 24 divisors.
Why 360 Has So Many Divisors
The number 360 is considered a highly composite number because it has more divisors than any smaller positive integer. Its prime factorization includes small prime numbers (2, 3, and 5) raised to relatively high powers, which allows for many combinations to form various divisors. This property makes 360 useful in various fields, such as geometry (360 degrees in a circle) and time measurement.
