what is the greatest common factor 42, 78, and 120?

To find the greatest common factor (GCF) of 42, 78, and 120, you can start by finding the factors of each number:

The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, and 42.
The factors of 78 are: 1, 2, 3, 6, 13, 26, 39, and 78.
The factors of 120 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120.

Now, we will look for the common factors among the three numbers:

The common factors of 42, 78, and 120 are: 1, 2, 3, and 6.

Therefore, the greatest common factor (GCF) of 42, 78, and 120 is 6.

To find the greatest common factor (GCF) of three numbers, such as 42, 78, and 120, we can use a method called prime factorization.

Step 1: Prime factorize each number:
- To prime factorize 42, we can start by dividing it by the smallest prime number, 2. 42 ÷ 2 = 21. Since 21 is not divisible by 2, we move to the next prime number, which is 3. 21 ÷ 3 = 7. Hence, the prime factorization of 42 is 2 × 3 × 7 = 2^1 × 3^1 × 7^1.
- To prime factorize 78, we start by dividing it by 2. 78 ÷ 2 = 39. Dividing 39 by 3 gives us 13, which is a prime number. Therefore, the prime factorization of 78 is 2 × 3 × 13 = 2^1 × 3^1 × 13^1.
- For 120, we divide it by 2 several times to get 60, 30, 15, and finally 5 (a prime number). Thus, the prime factorization of 120 is 2 × 2 × 2 × 3 × 5 = 2^3 × 3^1 × 5^1.

Step 2: Write down the common factors:
Looking at the prime factorizations, we can see that the common prime factors of 42, 78, and 120 are 2 (raised to the power of 1) and 3 (also raised to the power of 1). There are no other common prime factors.

Step 3: Multiply the common prime factors:
The GCF is obtained by multiplying the common prime factors: GCF(42, 78, 120) = 2^1 × 3^1 = 2 × 3 = 6.

Therefore, the greatest common factor of 42, 78, and 120 is 6.

Factors of 42 = 1, 42, 2, 21, 3, 14, 6, 7

Which of those numbers also go evenly into 78 and 120?