What is the LCM of 90 and 60?
Understand the Problem
The question is asking to find the least common multiple (LCM) of the numbers 90 and 60. To solve this, we will use the prime factorization method or the relationship between LCM and GCD (Greatest Common Divisor).
Answer
The LCM of 90 and 60 is $180$.
Answer for screen readers
The least common multiple (LCM) of 90 and 60 is $180$.
Steps to Solve
- Find the Prime Factorization of Each Number
Begin by finding the prime factors of both 90 and 60.
- The prime factorization of 90 is: $$ 90 = 2^1 \times 3^2 \times 5^1 $$
- The prime factorization of 60 is: $$ 60 = 2^2 \times 3^1 \times 5^1 $$
- Identify the Highest Powers of Each Prime Factor
Next, determine the highest powers of all the prime factors involved.
- For $2$, the highest power is $2^2$ (from 60).
- For $3$, the highest power is $3^2$ (from 90).
- For $5$, the highest power is $5^1$ (common in both).
- Multiply the Highest Powers Together
Now, multiply all the highest powers together to find the LCM. $$ \text{LCM} = 2^2 \times 3^2 \times 5^1 $$
- Calculate the LCM
Finally, calculate the result: $$ \text{LCM} = 4 \times 9 \times 5 $$
Calculating step-by-step:
- First, calculate $4 \times 9 = 36$.
- Then, calculate $36 \times 5 = 180$.
The least common multiple (LCM) of 90 and 60 is $180$.
More Information
The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both. It's often used in problems involving fractions, where finding a common denominator is necessary.
Tips
- A common mistake is to forget to include all prime factors or to miscalculate their powers. To avoid this, double-check your prime factorizations and ensure you have noted all the highest powers correctly.
