What is the LCM of 18 and 45?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 18 and 45. To find the LCM, we will identify the prime factors of each number and determine the smallest number that is divisible by both.
Answer
The least common multiple of 18 and 45 is $90$.
Answer for screen readers
The least common multiple of 18 and 45 is $90$.
Steps to Solve
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Find the prime factorization of 18
First, we'll factor 18 into its prime components.
The prime factors of 18 are:
$$ 18 = 2 \times 3^2 $$ -
Find the prime factorization of 45
Next, we'll factor 45 into its prime components.
The prime factors of 45 are:
$$ 45 = 3^2 \times 5 $$ -
Combine the prime factors
To find the LCM, we take the highest power of each prime factor from both numbers.
From the factorizations, we have:
- The highest power of 2 is $2^1$ (from 18)
- The highest power of 3 is $3^2$ (from either number)
- The highest power of 5 is $5^1$ (from 45)
Thus, the LCM can be calculated as:
$$ LCM = 2^1 \times 3^2 \times 5^1 $$
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Calculate the LCM
Now we will calculate the LCM using the combined prime factors:
$$ LCM = 2 \times 9 \times 5 $$
First, calculate $2 \times 9 = 18$, then multiply by 5:
$$ 18 \times 5 = 90 $$ -
Final LCM result
The least common multiple of 18 and 45 is therefore:
$$ LCM = 90 $$
The least common multiple of 18 and 45 is $90$.
More Information
The least common multiple (LCM) is useful in various applications such as finding common denominators in fractions or scheduling repeating events. The LCM of 18 and 45 being 90 indicates that both numbers are factors of 90, meaning any multiple of 90 can be divided by both.
Tips
- Forgetting to consider the highest power of each prime factor can lead to incorrect results when calculating the LCM.
- Mixing up the prime factorizations of the numbers may result in determining the wrong LCM. Always double-check the factorizations.
