Find the LCM of the given set of counting numbers.
$20,6,2$
Answer (1)
Find the prime factorization of each number: 20 = 2 2 × 5 , 6 = 2 × 3 , 2 = 2 .
Identify the highest power of each prime factor: 2 2 , 3 1 , 5 1 .
Multiply the highest powers together: LCM = 2 2 × 3 × 5 .
Calculate the LCM: 4 × 3 × 5 = 60 .
Explanation
Understanding the Problem We are asked to find the least common multiple (LCM) of the numbers 20, 6, and 2. The LCM is the smallest positive integer that is divisible by each of the given numbers.
Prime Factorization To find the LCM, we can use the prime factorization method. First, we find the prime factorization of each number:
20 = 2 2 × 5 6 = 2 × 3 2 = 2
Identifying Highest Powers Next, we identify the highest power of each prime factor present in the factorizations. The prime factors are 2, 3, and 5. The highest power of 2 is 2 2 , the highest power of 3 is 3 1 , and the highest power of 5 is 5 1 .
Calculating the LCM Finally, we multiply these highest powers together to get the LCM:
LCM = 2 2 × 3 × 5 = 4 × 3 × 5 = 60
Final Answer Therefore, the LCM of 20, 6, and 2 is 60.
Examples
In real life, the LCM is useful when scheduling events that occur at different intervals. For example, suppose you have three tasks: one that needs to be done every 20 days, another every 6 days, and a third every 2 days. The LCM of 20, 6, and 2 (which is 60) tells you that all three tasks will need to be done on the same day every 60 days. This helps you coordinate and plan your schedule efficiently.
