What is the least common multiple of 72 and 120?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 72 and 120. The LCM of two numbers is the smallest number that is a multiple of both numbers. To solve this, we can use the prime factorization method or the relationship with the greatest common divisor (GCD).
Answer
$360$
Answer for screen readers
The least common multiple of 72 and 120 is $360$.
Steps to Solve
- Find the prime factorization of each number
To find the least common multiple (LCM), we need the prime factorization of both numbers.
For 72: $$ 72 = 2^3 \times 3^2 $$
For 120: $$ 120 = 2^3 \times 3^1 \times 5^1 $$
- Identify the highest powers of all prime factors
Next, we determine the highest powers of each prime factor from both factorizations.
- For the prime factor 2, the highest power is $2^3$.
- For the prime factor 3, the highest power is $3^2$.
- For the prime factor 5, the highest power is $5^1$.
- Multiply the highest powers together
Now, we multiply these highest powers to find the LCM.
The LCM is: $$ LCM = 2^3 \times 3^2 \times 5^1 $$
- Calculate the value of the LCM
Now we calculate the LCM step by step:
First, calculate $2^3$: $$ 2^3 = 8 $$
Next, calculate $3^2$: $$ 3^2 = 9 $$
Now multiply these results together: $$ 8 \times 9 = 72 $$
Finally, multiply by $5^1$: $$ 72 \times 5 = 360 $$
Thus, the LCM of 72 and 120 is 360.
The least common multiple of 72 and 120 is $360$.
More Information
Finding the LCM is useful in many applications, such as solving problems involving fractions, where you need a common denominator. The method of prime factorization is a reliable approach for finding the LCM, as it ensures that all prime factors are included at their highest powers.
Tips
- One common mistake is forgetting to include all prime factors at their highest powers. Make sure to account for all primes from both numbers.
- Another mistake is incorrect multiplication of the prime factors when calculating the LCM. It's helpful to calculate in steps to avoid errors.
