What is the least common multiple of 6 and 18?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 6 and 18. To find the LCM, we can use the prime factorization method or list the multiples of each number and identify the smallest common one.
Answer
$18$
Answer for screen readers
The least common multiple (LCM) of 6 and 18 is 18.
Steps to Solve
- Find the prime factorization of each number
To find the LCM using prime factorization, start by determining the prime factors of the numbers.
For 6: The prime factorization is $6 = 2 \times 3$.
For 18: The prime factorization is $18 = 2 \times 3^2$.
- Identify the highest power of each prime factor
Next, identify the highest power of all prime factors involved.
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The prime factor 2 appears in both factorizations:
- In 6, it appears as $2^1$.
- In 18, it appears as $2^1$.
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The prime factor 3 also appears in both:
- In 6, it appears as $3^1$.
- In 18, it appears as $3^2$.
Thus we take the highest power for each prime factor:
- For 2, the highest power is $2^1$.
- For 3, the highest power is $3^2$.
- Calculate the LCM
Now, multiply the highest powers of each prime factor to find the LCM:
$$ LCM = 2^1 \times 3^2 $$
Calculating this gives:
$$ LCM = 2 \times 9 = 18 $$
The least common multiple (LCM) of 6 and 18 is 18.
More Information
The least common multiple is the smallest number that is a multiple of both original numbers. In this case, 18 is the smallest number that can be divided by both 6 and 18 without leaving a remainder. It's also important to note that when one number is a multiple of another, the larger number is often the LCM.
Tips
- Forgetting to list all prime factors: Always ensure to include all primes when factoring.
- Confusing LCM with GCD: Remember that LCM looks for the smallest common multiple, while GCD (greatest common divisor) looks for the largest factor they share.
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