Understanding Prime Factors and Least Common Multiples
The foundation of advanced mathematics relies heavily on understanding prime factors and least common multiples (LCM). When working with these concepts, students need to develop a systematic approach to break down numbers and find relationships between them.
Definition: Prime factors are the prime numbers that multiply together to make a number. A prime number can only be divided evenly by 1 and itself.
Let's explore how to express numbers as products of prime factors using factor trees. Taking the number 150 as an example, we can break it down systematically: First, divide 150 by 2 (giving 75), then break 75 into its factors (3 × 25), and finally break down 25 (5 × 5). This gives us 150 = 2 × 3 × 5². Writing numbers in this form helps identify common factors between numbers and simplifies calculations involving multiplication and division.
When finding the least common multiple (LCM) of numbers, listing multiples helps identify the smallest number that is divisible by both numbers. For instance, finding the LCM of 12 and 20: List the multiples of 12 (12, 24, 36, 48, 60) and 20 (20, 40, 60, 80) until you find the first common multiple. In this case, 60 is the LCM.
Example: To find the LCM of 12 and 20:
- Multiples of 12: 12, 24, 36, 48, 60
- Multiples of 20: 20, 40, 60, 80
- First common multiple: 60
Therefore, LCM(12,20) = 60