if the product of two numbers is 180 and the LCM of the two numbers is 60 then how do I determine what the GCF is?
Use formula
LCM x GCF = product of two numbers
GCF = 180 / 60= 3
To find the Greatest Common Factor (GCF) of two numbers, you need to consider their prime factors. Here's how you can determine the GCF given that the product of two numbers is 180 and their least common multiple (LCM) is 60:
Step 1: Find the prime factorization of both numbers.
Let's assume the two numbers are "x" and "y."
The prime factorization of the product (180) can be written as:
180 = 2 * 2 * 3 * 3 * 5 = 2^2 * 3^2 * 5
The prime factorization of the LCM (60) can be written as:
60 = 2 * 2 * 3 * 5 = 2^2 * 3 * 5
Step 2: The GCF is the product of the common prime factors, each raised to the minimum power.
In this case, the common prime factors are 2, 3, and 5 (since they exist in both the prime factorization of 180 and 60).
Let's determine the minimum power for each common prime factor:
For 2: The minimum power is 2 (since it appears as 2^2 in both factorizations).
For 3: The minimum power is 1 (since it appears as 3^1 in the factorization of 60 and 3^2 in the factorization of 180).
For 5: The minimum power is 1 (since it appears as 5^1 in both factorizations).
Now, multiply the common prime factors, each raised to the minimum power:
GCF = 2^2 * 3^1 * 5^1 = 4 * 3 * 5 = 60
Therefore, the GCF of the two numbers is 60.