lcm of 72 and 54
Understand the Problem
The question is asking to find the least common multiple (LCM) of the numbers 72 and 54. This will involve determining the multiples of both numbers and identifying the smallest multiple that they share.
Answer
$216$
Answer for screen readers
The least common multiple of 72 and 54 is $216$.
Steps to Solve
- Find the prime factorization of 72
To start, we need to break down 72 into its prime factors.
$$ 72 = 2^3 \times 3^2 $$
- Find the prime factorization of 54
Next, we do the same for 54.
$$ 54 = 2^1 \times 3^3 $$
- Identify the highest powers of each prime factor
Now, we find the highest powers of each prime factor present in both factorizations:
- For the prime factor 2: The highest power is $2^3$ (from 72).
- For the prime factor 3: The highest power is $3^3$ (from 54).
- Multiply the highest powers together
Now, we multiply these highest powers to find the LCM:
$$ LCM(72, 54) = 2^3 \times 3^3 $$
- Calculate the value
Now we perform the multiplication:
$$ LCM(72, 54) = 8 \times 27 = 216 $$
The least common multiple of 72 and 54 is $216$.
More Information
The least common multiple (LCM) is useful for problems involving fractions, scheduling, and finding common times in events.
Tips
- Mixing up the lowest common multiple with the greatest common divisor (GCD). Remember, LCM finds the smallest multiple, while GCD finds the largest divisor.
- Forgetting to use the highest powers of prime factors results in an incorrect LCM.
