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Prime factorization involves breaking down a number into its basic building blocks, the prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For instance, numbers like 2, 3, 5, and 7 are considered prime numbers. When you perform a prime factorization, you express a number as a product of these primes.

Let's look at how this helps with numbers such as 36, 54, and 162. By writing each of these numbers as a product of primes, we can see their components clearly:

• 36 becomes \(2^2 \times 3^2\).
• 54 becomes \(2 \times 3^3\).
• 162 becomes \(2 \times 3^4\).

With these expressions, you can easily identify prime numbers that each number shares, which is crucial when seeking the greatest common factor.

Common factors are numbers that appear as factors for each number in a set. When you have different numbers, each can be broken down into smaller numbers that multiply together to form the original number. For example, in the case of 36, 54, and 162, factors are those numbers we derive after their prime factorization.

For an easier approach to find common factors, compare the prime factorizations of each number to see which prime numbers they share:

• The prime factor 2 is present in all three numbers.
• The prime factor \(3^2\) (which is 9) is also common to all three.

These shared factors are what you use to calculate the greatest common factor; they are the prime numbers that each original number can be divided by evenly.

To fully understand greatest common factors, it's helpful to also consider the role of factors and multiples. A factor of a number is an integer that divides it evenly without leaving a remainder. For example, numbers like 1, 2, 3, 4, 6, 9, 12, 18, and so forth are all factors of 36 because they divide 36 without remainders.

Conversely, a multiple is the product of a number and any integer. For example, multiples of 3 include 6, 9, 12, 15, and continue infinitely. While factors help you decompose numbers, multiples help you understand the numbers they can build.

In the problem, when we compute the greatest common factor by multiplying the common factors \(2 \times 9\), we get 18. Hence, 18 is the largest number (or factor) that can divide 36, 54, and 162 without leaving any remainder.