Solving for X with LCM and HCF: A Case Study on 64, 80, and X
In this article, we explore the use of Least Common Multiple (LCM) and Highest Common Factor (HCF) to solve for an unknown value X. Specifically, we examine the scenario where the LCM of 64, 80, and X is 960, and the HCF is 16. By understanding and applying the relationship between LCM, HCF, and the numbers involved, we will derive the possible values of X.
Understanding the Relationship Between LCM and HCF
The relationship between LCM, HCF, and the numbers a, b, and c is given by:
LCM(a, b, c) * HCF(a, b, c) a * b * c
Step-by-Step Solution
Step 1: Understanding the RelationshipGiven the numbers 64, 80, and X, we know the LCM is 960 and the HCF is 16.
Step 2: Calculate the Product of the NumbersWe start by substituting the known values into the relationship:
960 * 16 64 * 80 * X
Calculating the left side:
960 * 16 15360
Calculating the product of 64 and 80:
64 * 80 5120
Step 3: Set Up the EquationNow we can set up the equation to solve for X:
15360 5120 * X
Step 4: Solve for XRearrange the equation to solve for X:
X 15360 / 5120
Calculating the right side:
X 3
Step 5: Verify HCF and LCMFinally, we verify if X 3 gives the correct HCF and LCM with 64 and 80.
HCF Calculation The prime factorization of 64 is 2^6. The prime factorization of 80 is 2^4 * 5^1. The prime factorization of 3 is 3^1.The only common prime factor is 2:
HCF(64, 80, 3) 2^4 16
LCM Calculation The LCM takes the highest power of each prime factor: For 2: 2^6 For 5: 5^1 For 3: 3^1Thus, LCM(64, 80, 3) 2^6 * 5^1 * 3^1 64 * 5 * 3 960
Both calculations confirm that the HCF is 16 and the LCM is 960.
Conclusion
The possible value of X is: boxed{3}
Exploring Further
Given that the biggest multiple of the three numbers can be is 960, and both LCM and HCF are distributive, the LCM of 64 and 80 is 320. A loop can be set up to examine each X from 1 to 960, checking whether it is the LCM of 320 and X. If this is so then we can check that the three-number HCF is 16.