The GCF of two numbers is 15. Their LCM is 450. What are the two numbers? (Enter your answers as a comma-separated list.)
To find the two numbers given their greatest common factor (GCF) and least common multiple (LCM), we can use the formula:
LCM × GCF = product of the two numbers
In this case, the GCF is 15 and the LCM is 450. So, using the formula, we can write:
450 × 15 = product of the two numbers
By multiplying 450 and 15, we find that the product of the two numbers is 6750.
Now, let's find the two numbers by using the product we just calculated. We need to find two numbers whose product is 6750 and whose GCF is 15.
Factorizing 6750, we get:
6750 = 2 × 3 × 3 × 5 × 5 × 3
To find the two numbers with a GCF of 15, we need to split these prime factors into two sets. One set will have the common prime factors (together with their exponents) and the other set will have the remaining prime factors.
The common prime factors are:
3 × 5 × 5
And the additional prime factors are:
2 × 3
Using these sets of prime factors, we can now find the two numbers.
The first number will have the common prime factors and the second number will have the additional prime factors:
First number: 3 × 5 × 5 = 75
Second number: 2 × 3 = 6
So, the two numbers with a GCF of 15 and an LCM of 450 are 75 and 6.
Therefore, the answer is "75, 6".