To write 360 as a product of its prime factors, you express it as 2 × 2 × 2 × 3 × 3 × 5, or in exponential form, 2³ × 3² × 5¹.
Prime factorization is the process of breaking down a composite number into its prime number components. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11). This fundamental concept is crucial in various mathematical operations, from simplifying fractions to finding the greatest common divisor and least common multiple.
Understanding Prime Factorization
Every composite number has a unique set of prime factors, as stated by the Fundamental Theorem of Arithmetic. Finding these factors involves systematically dividing the number by the smallest possible prime numbers until only prime numbers remain.
Step-by-Step Prime Factorization of 360
Let's break down 360 into its prime factors using the division method:
- Start with the smallest prime number, 2:
- 360 ÷ 2 = 180
- Continue dividing by 2 if possible:
- 180 ÷ 2 = 90
- 90 ÷ 2 = 45
(Since 45 is not divisible by 2, we move to the next prime number.)
- Divide by the next prime number, 3:
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
(Since 5 is not divisible by 3, we move to the next prime number.)
- Divide by the next prime number, 5:
- 5 ÷ 5 = 1
(We stop when we reach 1.)
- 5 ÷ 5 = 1
By following these steps, we find that the prime factors are 2, 2, 2, 3, 3, and 5.
Summary Table of Prime Factorization
| Step | Number | Prime Factor | Result |
|---|---|---|---|
| 1 | 360 | 2 | 180 |
| 2 | 180 | 2 | 90 |
| 3 | 90 | 2 | 45 |
| 4 | 45 | 3 | 15 |
| 5 | 15 | 3 | 5 |
| 6 | 5 | 5 | 1 |
Thus, 360 can be written as the product of its prime factors: 2 × 2 × 2 × 3 × 3 × 5.
Exponential Form
For a more concise representation, we can express the repeated factors using exponents:
- The prime factor 2 appears 3 times, so it's 2³.
- The prime factor 3 appears 2 times, so it's 3².
- The prime factor 5 appears 1 time, so it's 5¹.
Therefore, the prime factorization of 360 in exponential form is 2³ × 3² × 5¹.
Practical Applications of Prime Factorization
Prime factorization isn't just a mathematical exercise; it has several practical uses:
- Finding the Greatest Common Divisor (GCD): To find the largest number that divides two or more numbers evenly.
- Finding the Least Common Multiple (LCM): To find the smallest number that is a multiple of two or more numbers.
- Simplifying Fractions: By factoring the numerator and denominator into primes, common factors can be canceled.
- Cryptography: Prime numbers play a fundamental role in modern encryption methods, ensuring secure communication.
Understanding how to break down numbers into their prime components is a foundational skill in number theory, offering insights into the structure and properties of integers.
