What is the lcm of 8 and 24?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 8 and 24. To find the LCM, we can list the multiples of each number and identify the smallest multiple they have in common, or use the prime factorization method.
Answer
$24$
Answer for screen readers
The least common multiple of 8 and 24 is $24$.
Steps to Solve
- List the multiples of each number
Start by listing the first few multiples of both 8 and 24.
- Multiples of 8: 8, 16, 24, 32, 40, 48, 56, ...
- Multiples of 24: 24, 48, 72, 96, ...
- Identify the common multiples
Next, identify the common multiples from the lists made above. The common multiples of 8 and 24 are:
- Common multiples: 24, 48, ...
- Find the least common multiple (LCM)
The least common multiple is the smallest number that appears in both lists of multiples. From the list, the smallest common multiple is:
$$ LCM(8, 24) = 24 $$
Alternatively, we can find the LCM using prime factorization:
- Prime factorization approach
Now we can also use prime factorization for each number:
- Prime factorization of 8: $8 = 2^3$
- Prime factorization of 24: $24 = 2^3 \times 3^1$
To find the LCM, take the highest power of each prime factor:
- From both, we take $2^3$ and $3^1$.
So the LCM can also be calculated as:
$$ LCM(8, 24) = 2^3 \times 3^1 = 8 \times 3 = 24 $$
The least common multiple of 8 and 24 is $24$.
More Information
The least common multiple is important for solving problems involving multiples, particularly in fractions, as it helps in finding a common denominator. In this case, both methods (listing multiples and using prime factorization) led to the same result.
Tips
- Failing to list enough multiples to find the least common multiple.
- Confusing LCM with the greatest common divisor (GCD), which is a different concept.