Introduction
Understanding how to find the least common multiple (LCM) is a fundamental skill in mathematics, particularly useful in various applications such as simplifying fractions, solving problems involving fractions, and more. In this article, we will explore the process of determining the smallest number that is divisible by both 4 and 9, which is essentially finding the LCM of 4 and 9.
What is the Least Common Multiple (LCM)?
The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of the integers. In simpler terms, it is the smallest number that is a common multiple of the given numbers.
Prime Factorization
To find the LCM of 4 and 9, we first express each number in its prime factorized form:
Prime Factorization of 4
4 2^2
Prime Factorization of 9
9 3^2
Calculating the LCM
Once we have the prime factorization, we determine the LCM by taking the highest powers of all prime factors involved.
Steps to Determine the LCM
Identify the prime factors involved: 2 and 3. Take the highest power of 2, which is 2^2. Take the highest power of 3, which is 3^2. Multiply these highest powers together:LCM 2^2 times; 3^2 4 times; 9 36
Alternative Approaches
There are several methods to determine the LCM, and we will explore a few more to ensure a comprehensive understanding.
Approach 1: Listing Multiples
Another common method is to list the multiples of each number and find the smallest number that appears in both lists.
Multiples of 4
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144,...
Multiples of 9
9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144,...
The smallest number that is common in both lists is 36.
Approach 2: Prime Factorization Directly
A more straightforward method is to directly use the prime factorization to find the LCM. We know:
4 2^2
9 3^2
Therefore, the LCM is:
LCM 2^2 times; 3^2 4 times; 9 36
Conclusion
By following these steps and methods, we have determined that the smallest number divisible by both 4 and 9 is 36. Understanding the LCM is crucial for solving various mathematical problems, and the methods discussed here provide a solid foundation for this important concept.