Finding Pairs of Numbers with a Specific LCM
Today, we will explore how to find pairs of numbers a and b such that their least common multiple (LCM) is 40. We will use the prime factorization of 40 and the properties of LCM to derive these pairs. Let's delve into the steps involved.
"Understanding the Prime Factorization of 40
First, we need to understand the prime factorization of 40. The prime factorization of 40 is:
40 23 times; 51
For two numbers to have an LCM of 40, their combined prime factors must be able to form 23 and 51. This means that one of the numbers must have 5 as a factor, and the other must have either 23 or 8 as a factor. We can express this as:
a 5a and b 23, so pairs could be (5, 8) a 23 and b 5a, so pairs could be (8, 5) a 5a and b 2b, where A 1 or 0, B 0 or 1 or 2 or 3. Possible pairs include (5, 2), (5, 4), (5, 8), (10, 2), (10, 4), (20, 2), (20, 4), (40, 1), (40, 2), (40, 4), (40, 8), (40, 5), (40, 10), (40, 20), (40, 40)Note that some of these pairs are duplicates and should be removed. Let's remove duplicates and list the unique pairs:
5, 8 8, 5 5, 4 5, 2 10, 2 10, 4 20, 2 20, 4 40, 1 40, 2 40, 4 40, 8 40, 10 40, 20 40, 40Systematic Case Analysis
Now, let's systematically analyze the cases based on the highest common value.
Case 1: Max(a, b) 40
Here, the LCM of 40 with any divisor of 40 is 40. The divisors of 40 are 1, 2, 4, 5, 8, 10, 20, 40. Therefore, this case gives eight solutions:
1, 40 2, 40 4, 40 5, 40 8, 40 10, 40 20, 40 40, 40Case 2: Max(a, b) 20
Here, the LCM of 20 with any divisor of 20 is 20, not 40. The divisors of 20 are 1, 2, 4, 5, 10, 20. Therefore, this case gives one solution:
20, 8Case 3: Max(a, b) 10
Here, the LCM of 10 with any divisor of 10 is 10, not 40. The divisors of 10 are 1, 2, 5, 10. Therefore, this case gives one solution:
8, 10Case 4: Max(a, b) 8
Here, the LCM of 8 with any non-factor of 8 is 40. The non-factors of 8 are 5, 10, 15, 20, 25, 30, 35, 40. Therefore, this case gives one solution:
8, 5Case 5: Max(a, b)
Here, the product of a and b is less than 40, hence no more solutions.
Summary of All Solutions
In summary, the pairs of numbers (a, b) such that their LCM is 40 are:
(1, 40) (2, 40) (4, 40) (5, 40) (8, 40) (10, 40) (20, 40) (40, 40) (20, 8) (8, 10) (8, 5)This systematic approach ensures that we cover all possible pairs and eliminate any duplicates.