The Relationship Between Product, GCD, and LCM
The relationship between the product of two numbers and their least common multiple (LCM) is given by the equation:
- Product of two numbers = LCM × GCD
This means that if we know the product of two numbers and their greatest common divisor (GCD), we can calculate their LCM.
Understanding the Given Product
- The product of the two numbers is 840.
- To find the possible LCM values, we first need to determine the GCD.
Finding the GCD
- The GCD of two numbers can vary based on the numbers chosen.
- For example, if the two numbers are factors of 840, their GCD can range from 1 to the number itself.
Calculating Possible LCMs
- Since LCM can vary based on the GCD, we can use the formula to find possible LCMs:
- If GCD = 1 (co-prime numbers):
LCM = 840 / 1 = 840
- If GCD = 2:
LCM = 840 / 2 = 420
- If GCD = 3:
LCM = 840 / 3 = 280
- If GCD = 7:
LCM = 840 / 7 = 120
- If GCD = 12:
LCM = 840 / 12 = 70
- Continuing this way, we can find different possible values for the LCM depending on the GCD.
Conclusion
- The LCM of two numbers whose product is 840 can be various values, including 840, 420, 280, and others, depending on the GCD.
- Thus, the possible LCMs can be determined by analyzing the factors of 840 and their GCDs.