common factors
To understand common factors, let’s first define what a 'factor' is. A factor of a number is an integer that can be multiplied by another integer to produce that number. For example, 3 is a factor of 12 because 3 × 4 = 12. Now, for common factors, we look at two or more numbers and identify the factors they share. In our case with 144 and 54, we list the factors individually:
- Factors of 144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
- Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
When comparing both lists, the numbers that appear in both are common factors. Hence, the common factors of 144 and 54 are: 1, 2, 3, 6, 9, 18.
relatively prime numbers
Two numbers are considered relatively prime, or coprime, if their greatest common factor (GCF) is 1. This means that 1 is the only number that divides both of them without leaving a remainder. For example, consider the numbers 15 and 28.
- Factors of 15: 1, 3, 5, 15
- Factors of 28: 1, 2, 4, 7, 14, 28
The only common factor is 1, so 15 and 28 are relatively prime. In contrast, let’s look at our original numbers, 144 and 54. We determined the GCF to be 18, which is not 1. Therefore, 144 and 54 are not relatively prime.
factorization
Factorization is the process of breaking down a number into its constituent factors, which multiply together to give the original number. There are different types of factorization, but one of the most commonly used methods is prime factorization. Prime factorization involves expressing a number as a product of its prime factors.
- For example, the prime factors of 144 are 2 and 3: \[144 = 2^4 \times 3^2\]
- The prime factors of 54 are also 2 and 3: \[54 = 2 \times 3^3\]
Prime factorization is particularly useful because it helps in identifying common factors and calculating the greatest common factor (GCF). We can see that both 144 and 54 share the prime factors 2 and 3, which leads us to the common factors identified earlier.
