The LCM of two numbers is 120. The GCF is 6. One number is 6 more than the other. What are the numbers. Can someone please explain how to get the answer?

To find the two numbers, let's start by finding the prime factorization of 120, since the LCM is given as 120:

Prime factorization of 120:
120 = 2^3 × 3 × 5

Now, let's find the prime factorization of the GCF, which is 6:

Prime factorization of 6:
6 = 2 × 3

The GCF has a 2 and a 3 as factors. Since the LCM has those same factors, we can conclude that the two numbers have the same prime factors, except for any additional factors they might have.

Next, we know that one number is 6 more than the other. Let's assume one number is "x" and the other number is "x + 6".

To find the LCM of these two numbers, we need to determine their prime factorization. Since we already know the factors are 2 and 3, we can focus on finding the additional factors for each number.

For "x", we need to find the factors that are not already accounted for in the GCF (2 and 3). The remaining factors of the prime factorization of 120 are 2^3 × 5. So, "x" has a factor of 5.

For "x + 6", we can apply the same process. The remaining factors of the prime factorization of 120 are 2^3 × 5. So, "x + 6" also has a factor of 5.

We have found all the factors for both numbers: "x" has factors of 2, 3, and 5; "x + 6" has factors of 2, 3, and 5.

To find the numbers, we need to determine the value of "x" that satisfies the given conditions. Since "x" and "x + 6" have the same factors, we can conclude that the two numbers must be multiples of their common factors. In this case, the factors are 2, 3, and 5.

The highest power of 2 in 120 is 2^3, so the multiples of 2 would be 2, 4, 8, 16, 32, 64, and 120.

The highest power of 3 in 120 is 3, so the multiples of 3 would be 3, 6, 9, 12, 15, 18, and 120.

The multiples of 5 would be 5, 10, 15, ..., but since we already found that both numbers have a factor of 5, we only need to consider the values that are multiples of 5 after adding 6. So, the multiples of 5 in this case are 11, 16, 21, and so on.

To satisfy the condition that one number is 6 more than the other, we can find the pair of numbers where one number is 6 greater than the other.

Looking at the list, we find that 15 and 21 satisfy the condition. These numbers have the prime factors 2, 3, and 5, and one is indeed 6 more than the other.

Therefore, the two numbers are 15 and 21.