The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the given numbers. The Greatest Common Divisor (GCD) of two or more numbers is the largest number that divides all the given numbers without leaving a remainder....
The Relationship Between LCM, GCD, and Numbers
There is a fundamental relationship between the LCM, GCD, and the numbers themselves. This relationship is expressed by the following equation:
LCM(a, b) * GCD(a, b) = a * b
This equation holds true for any two positive integers 'a' and 'b'. We can extend this concept to three numbers, 'a', 'b', and 'c', as follows:
LCM(a, b, c) * GCD(a, b, c) = a * b * c
Applying the Relationship to the Problem
In our problem, we are given the following information:
- LCM = 7920
- GCD = 12
- a = 48
- b = 264
- c = unknown (with a factor of 9)
We can use the extended relationship equation to find the unknown number 'c':
7920 * 12 = 48 * 264 * c
Solving for 'c', we get:
c = (7920 * 12) / (48 * 264) = 9
Therefore, the third number is 9.
Factor Notation and Finding the Third Number
Since we know that the third number has a factor of 9, we can use factor notation to find the complete factorization of the third number. We can start by finding the prime factorization of each of the given numbers and the LCM and GCD:
- 48 = 24 * 31
- 264 = 23 * 31 * 111
- 7920 = 24 * 32 * 51 * 111
- 12 = 22 * 31
To find the third number, we can follow these steps:
- **Identify the common factors:** The common factors of all the numbers (including the unknown number) are 22 and 31. These represent the GCD.
- **Find the highest powers in the LCM:** The LCM includes the highest powers of all the prime factors. In this case, the highest power of 2 is 24, of 3 is 32, of 5 is 51, and of 11 is 111.
- **Find the missing factor:** Since we know that one factor of the third number is 9 (32), we need to find the other factors that are missing from the GCD but present in the LCM. These are 22 * 51 * 111.
- **Multiply to get the third number:** Multiplying the missing factors together, we get: 22 * 51 * 111 * 32 = 1980.
Therefore, the third number is 1980. We can verify this by using the factor notation for the third number: 1980 = 22 * 32 * 51 * 111.
Conclusion
By understanding the relationship between the LCM, GCD, and the numbers themselves, we can efficiently find the unknown number in a given set. Using factor notation allows us to break down the numbers into their prime factors and systematically find the missing factor to determine the unknown number.