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Divisibility is a concept in mathematics that helps us determine if one number can be divided by another without leaving a remainder. It’s essential for finding factors and prime factors.
Take the number 14, for example. To find its factors, we need to check if it’s divisible by smaller numbers.
Start with 2, the smallest prime number. Since 14 is even, it's divisible by 2. We can say that 14 divided by 2 equals 7, which means 2 is one of its factors.
Understanding divisibility rules for other numbers can also help:

• If a number ends in 0 or 5, it’s divisible by 5.
• If the sum of a number’s digits is a multiple of 3, it’s divisible by 3.
• If a number ends in 0, it’s divisible by 10.

These rules make it easier to break down larger numbers into their factors.

Prime numbers play a critical role in factorization. A prime number is a number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Essentially, it has only two distinct positive divisors: 1 and itself.
Examples of prime numbers include 2, 3, 5, 7, 11, and so on. They're the building blocks for all natural numbers, as every number can be expressed as a product of primes.
In our example with the number 14, we saw that 2 is a prime number. The number we got after dividing 14 by 2 is 7, which is also a prime number. So, 14 can be split into the prime factors 2 and 7.
Recognizing prime numbers quickly can simplify many math problems, especially when dealing with prime factorization.

Factorization is the process of breaking down a number into its product of factors. In prime factorization, we aim to express a number as a product of prime numbers.
The steps for prime factorization are simple:
1. Start with the smallest prime number, which is 2, and check if the given number is divisible by it.
2. If yes, divide the number by 2 and repeat the process with the quotient until it’s no longer divisible by 2.
3. Move to the next smallest prime number and repeat the process.
For example, let's factorize 14:
1. Check if it's divisible by 2. It is, because 14 is even.
2. Divide 14 by 2 to get 7.
3. Check if 7 is a prime number. It is, because it has no divisors other than 1 and itself.
Therefore, the prime factorization of 14 is: \( 14 = 2 \times 7 \).