What is the least common multiple of 36 and 45?
Understand the Problem
The question is asking to find the least common multiple (LCM) of the numbers 36 and 45. To solve this, we will identify the prime factors of both numbers and then use those to calculate the LCM.
Answer
180
Answer for screen readers
The least common multiple (LCM) of 36 and 45 is 180.
Steps to Solve
- Find the prime factorization of each number
First, we'll break down both 36 and 45 into their prime factors.
For 36:
- 36 can be divided by 2: $$ 36 = 2 \times 18 $$
- Then divide 18 by 2: $$ 18 = 2 \times 9 $$
- Finally, divide 9 by 3: $$ 9 = 3 \times 3 $$ Thus, the prime factorization of 36 is: $$ 36 = 2^2 \times 3^2 $$
For 45:
- 45 can be divided by 3: $$ 45 = 3 \times 15 $$
- Then divide 15 by 3: $$ 15 = 3 \times 5 $$ Thus, the prime factorization of 45 is: $$ 45 = 3^2 \times 5^1 $$
- Determine the highest powers of each prime factor
Next, we'll take note of the highest powers of all prime factors from both factorizations.
- The prime factors we have are 2, 3, and 5.
- The highest power of 2 is $2^2$ (from 36).
- The highest power of 3 is $3^2$ (common to both).
- The highest power of 5 is $5^1$ (from 45).
- Calculate the LCM using these prime factors
Now, we can calculate the LCM by multiplying the highest powers of each prime factor together: $$ \text{LCM} = 2^2 \times 3^2 \times 5^1 $$
Calculating that gives us: $$ \text{LCM} = 4 \times 9 \times 5 $$ First, calculate $4 \times 9 = 36$.
Then, multiply $36 \times 5 = 180$.
- Conclusion with the LCM of the two numbers
Thus, the least common multiple (LCM) of 36 and 45 is: $$ \text{LCM} = 180 $$
The least common multiple (LCM) of 36 and 45 is 180.
More Information
The LCM of two numbers is the smallest number that is evenly divisible by both. In this case, 180 can be reached by multiplying the highest prime factors of each number. This concept is commonly used in problems involving fractions, where you need to find a common denominator.
Tips
- Not finding the correct prime factorization: Ensure each number is broken down into only prime numbers.
- Forgetting to take the highest power for each prime factor: Always check both numbers and take the maximum exponent.
- Miscalculating the final multiplication step: Double-check arithmetic to avoid mistakes.