lcm of 54 and 72
Understand the Problem
The question is asking for the least common multiple (LCM) of the two numbers, 54 and 72. To find the LCM, we will determine the prime factorization of each number and then use those factors to calculate the LCM.
Answer
The least common multiple (LCM) of 54 and 72 is $216$.
Answer for screen readers
The least common multiple (LCM) of 54 and 72 is $216$.
Steps to Solve
- Prime Factorization of 54 We start by finding the prime factors of 54.
54 can be factored as: $$ 54 = 2 \times 27 $$ Next, 27 can be broken down into: $$ 27 = 3 \times 9 $$ Continuing, we factor 9: $$ 9 = 3 \times 3 $$ Thus, the complete prime factorization of 54 is: $$ 54 = 2^1 \times 3^3 $$
- Prime Factorization of 72 Now, we find the prime factors of 72.
72 can be factored as: $$ 72 = 8 \times 9 $$ Breaking down each component gives us: $$ 8 = 2^3 $$ $$ 9 = 3^2 $$ Combining these, we get the prime factorization of 72 as: $$ 72 = 2^3 \times 3^2 $$
- Determine the LCM To find the LCM, we take the highest power of each prime factor from the factorizations.
From the factorizations we have:
- The prime factor 2 appears as $2^1$ in 54 and $2^3$ in 72. We take $2^3$.
- The prime factor 3 appears as $3^3$ in 54 and $3^2$ in 72. We take $3^3$.
Thus, the LCM is calculated as: $$ LCM = 2^3 \times 3^3 $$
- Calculate the LCM Now we perform the multiplication: $$ LCM = 8 \times 27 $$ Calculating this gives: $$ 8 \times 27 = 216 $$
The least common multiple (LCM) of 54 and 72 is $216$.
More Information
Finding the least common multiple is useful in various applications, such as adding fractions with different denominators or scheduling events that occur at different intervals. The LCM helps identify a common timeframe.
Tips
Some common mistakes include:
- Not fully breaking down each number into its prime factors.
- Selecting the wrong powers of prime factors when determining the LCM.
To avoid these errors, always ensure you have the complete prime factorization and carefully select the highest powers.
