What is the LCM of 72 and 120?
Understand the Problem
The question is asking to find the least common multiple (LCM) of the numbers 72 and 120. To solve this, we can use the prime factorization of each number or the relationship between the GCF and LCM.
Answer
$360$
Answer for screen readers
The least common multiple (LCM) of 72 and 120 is $360$.
Steps to Solve
- Prime Factorization of 72
To find the prime factorization of 72, we divide 72 by the smallest prime numbers until we reach 1.
$$
72 = 2 \times 36
$$
$$
36 = 2 \times 18
$$
$$
18 = 2 \times 9
$$
$$
9 = 3 \times 3
$$
So, the prime factorization of 72 is: $$ 72 = 2^3 \times 3^2 $$
- Prime Factorization of 120
Next, we do the same for 120.
$$
120 = 2 \times 60
$$
$$
60 = 2 \times 30
$$
$$
30 = 2 \times 15
$$
$$
15 = 3 \times 5
$$
Thus, the prime factorization of 120 is: $$ 120 = 2^3 \times 3^1 \times 5^1 $$
- Finding the LCM
To find the least common multiple, we take the highest power of each prime factor from both factorizations.
- From $2$: maximum is $2^3$
- From $3$: maximum is $3^2$
- From $5$: maximum is $5^1$
Combining these, we get: $$ LCM = 2^3 \times 3^2 \times 5^1 $$
Calculating this step by step:
- $2^3 = 8$
- $3^2 = 9$
- $5^1 = 5$
- Final LCM Calculation
Now, multiply these values together: $$ LCM = 8 \times 9 \times 5 $$
Calculating:
- First, $8 \times 9 = 72$
- Then, $72 \times 5 = 360$
Thus: $$ LCM(72, 120) = 360 $$
The least common multiple (LCM) of 72 and 120 is $360$.
More Information
The least common multiple is important in various applications, such as finding common denominators in fractions and scheduling events that repeat over different intervals. The LCM can be found using different methods such as prime factorization or using the relationship with the greatest common factor (GCF).
Tips
- Ignoring higher powers: When finding the LCM, it's essential to select the highest power of each prime factor. Failing to do this can lead to an incorrect LCM.
- Confusing LCM with GCF: Make sure to distinguish between LCM (least common multiple) and GCF (greatest common factor), as they serve different purposes in problems.
