91Ó°ÊÓ

When we talk about prime numbers, we refer to numbers that have exactly two distinct positive divisors: 1 and the number itself. In other words, a prime number cannot be divided evenly by any other number except for 1 and itself.
Some basic examples of prime numbers are 2, 3, 5, 7, and 11. Notice that 2 is the only even prime number. All other even numbers can be divided by 2, so they're not prime.
Prime numbers are like building blocks in mathematics. When we perform prime factorization, we break down composite numbers (numbers that are not prime) into products of prime numbers.
Understanding prime numbers is important for prime factorization. They help us simplify and manipulate larger numbers more easily.

Divisibility rules are shortcuts that help us determine if one number is divisible by another without performing division.
Here are some useful divisibility rules:
1. **Divisibility by 2:** A number is divisible by 2 if it's even (it ends in 0, 2, 4, 6, or 8).
2. **Divisibility by 3:** If the sum of the digits of the number is divisible by 3, then the number itself is also divisible by 3.
3. **Divisibility by 5:** A number is divisible by 5 if it ends in 0 or 5.
In the exercise provided, these rules were used to factorize the number 120. Recognizing that it is even, we divided it by 2 multiple times until no longer possible, then used the rule for 3 when checking divisibility for the number 15.

Let's break down the factorization process we followed for the number 120:
1. **Step 1:** **Divisibility by 2** - Since 120 is even, divide by 2 to get 60.
2. **Step 2:** **Divisibility by 2** - 60 is still even, divide by 2 to get 30.
3. **Step 3:** **Divisibility by 2** - 30 is also even, divide by 2 to get 15.
4. **Step 4:** **Divisibility by 3** - 15 is not even. Checking the sum of the digits (1 + 5 = 6), which is divisible by 3, divide by 3 to get 5.
5. **Step 5:** **Prime Number** - 5 is only divisible by 1 and itself. Thus, we have reached a prime number.
The prime factorization of 120 is thus the product of the divisors we found: $$120 = 2^3 * 3 * 5$$
These steps help ensure we methodically identify and divide out prime factors, ensuring an accurate factorization of the number.