What is the least common multiple of 54 and 72?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 54 and 72. To find the LCM, we can use the prime factorization method or a listing method to identify the smallest multiple that both numbers share.
Answer
The least common multiple of 54 and 72 is $216$.
Answer for screen readers
The least common multiple (LCM) of 54 and 72 is $216$.
Steps to Solve
- Find the prime factorization of each number
Start by finding the prime factors of 54 and 72.
For 54, we can factor it as: $$ 54 = 2 \times 3^3 $$
For 72, we get: $$ 72 = 2^3 \times 3^2 $$
- Identify the highest power of each prime factor
Next, we need to identify the highest power of each prime factor from both factorizations.
From our 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).
- Calculate the LCM using the highest powers
Now we can calculate the LCM by multiplying these highest powers together: $$ LCM = 2^3 \times 3^3 $$
Calculate the values: $$ 2^3 = 8 $$ $$ 3^3 = 27 $$
Therefore, $$ LCM = 8 \times 27 $$
- Find the final result
Now, multiply to find the LCM: $$ LCM = 216 $$
The least common multiple (LCM) of 54 and 72 is $216$.
More Information
The least common multiple (LCM) is helpful in various applications, such as solving problems with fractions, scheduling events, or finding common denominators. The prime factorization method is one of the most efficient ways to find the LCM.
Tips
- Forgetting to include the highest powers of each prime factor when calculating the LCM.
- Incorrectly performing multiplication when calculating the final LCM value.
