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To understand factors, think of them as the basic building blocks of a number. A factor is a number that divides another number completely, leaving no remainder behind. For example, when we say 3 is a factor of 12, we mean that 12 can be divided by 3 with no leftover fraction.

• Factors of a number always include 1 and the number itself, because any number is divisible by these.
• Every number has a different set of factors that multiply together to give the original number.

To find the factors of a number, you can start from 1 and work your way up. Each time you find a pair of numbers that multiply to give the original number, you've found two factors. For instance, 12 has factors like 1, 2, 3, 4, 6, and 12. You can stop when you reach the square root of the number, as factors repeat beyond that point.

Once we have the list of factors for multiple numbers, we look for common elements across these lists. These are known as common factors. Common factors are numbers that are factors of all the given numbers.
For instance, let's consider the numbers 12, 18, and 20:

• Factors of 12 are 1, 2, 3, 4, 6, and 12.
• Factors of 18 are 1, 2, 3, 6, 9, and 18.
• Factors of 20 are 1, 2, 4, 5, 10, and 20.

By carefully checking, we notice that 1 and 2 appear in all three lists. That means 1 and 2 are common factors of 12, 18, and 20.
Identifying common factors helps in simplifying problems and finding the greatest common factor, which is the largest of these shared numbers.

Prime factorization takes us a step further by breaking numbers down into their prime factors, which are factors that are only divisible by 1 and themselves. Prime numbers are the building blocks for all numbers, similar to factors, but more specialized.
To perform prime factorization, we repeatedly divide the number by the smallest possible prime numbers, starting from 2. When a number is no longer divisible by a prime, move to the next larger prime.
Consider the number 18:

• Divide by 2: 18 ÷ 2 = 9.
• 9 is not divisible by 2, so move to next prime which is 3: 9 ÷ 3 = 3.
• 3 is a prime number and divides itself: 3 ÷ 3 = 1.

Thus, the prime factorization of 18 is 2 × 3 × 3.
This process aids in understanding the internal structure of a number and is helpful in determining the greatest common factor as it allows comparison of prime factors across different numbers.