The greatest common factor (GCF), also known as the greatest common divisor (GCD), of a set of numbers is the largest number that divides all the numbers in the set without leaving a remainder.
To find the GCF of 42, 126, and 210, we can use two methods: prime factorization and the listing of factors. Let’s go through both approaches:
Method 1: Prime Factorization
1. **Find the prime factorization of each number:**
- 42 can be factored into prime numbers as follows: 42 = 2 × 3 × 7
- 126 can be factored into prime numbers as follows: 126 = 2 × 3 × 3 × 7, which can also be written as 2 × 3² × 7
- 210 can be factored into prime numbers as follows: 210 = 2 × 3 × 5 × 7
2. **Identify the common prime factors:** The common prime factors from the factorizations above are 2, 3, and 7.
3. **Select the lowest power of each common factor:**
- 2 appears in all three factorizations with a power of 1.
- 3 appears in all factorizations, and the lowest power is 1 (2 appears only once in 42, once in 126, and once in 210).
- 7 appears in all of them with a power of 1.
4. **Multiply these together to get the GCF:**
GCF = 21 × 31 × 71 = 2 × 3 × 7 = 42
Method 2: Listing Factors
1. **List the factors of each number:**
- Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
- Factors of 126: 1, 2, 3, 6, 7, 9, 14, 21, 42, 63, 126
- Factors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210
2. **Identify the common factors:** The common factors are 1, 2, 3, 6, 7, 14, 21, and 42.
3. **Select the greatest of these common factors:** The greatest common factor is 42.
Conclusion
Therefore, the greatest common factor of 42, 126, and 210 is 42.
This means that 42 is the largest number that can evenly divide each of the numbers without leaving a remainder.
So the GCF of 42, 126, and 210 is 42.