What is the lcm 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 can use the prime factorization method or list the multiples of each number and find the smallest one that is common to both lists.
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
To start, we need to break down each number into its prime factors.
For 36: $$ 36 = 2^2 \times 3^2 $$
For 45: $$ 45 = 3^2 \times 5^1 $$
- Identify the highest power of each prime factor
Next, we list all the prime factors that appear in the factorizations. We take the highest power of each prime factor from both numbers.
- For the prime factor $2$: the highest power from 36 is $2^2$.
- For the prime factor $3$: the highest power from both is $3^2$.
- For the prime factor $5$: the highest power from 45 is $5^1$.
- Multiply the highest powers together
Now, we take the highest power of each prime factor and multiply them to find the LCM.
$$ LCM = 2^2 \times 3^2 \times 5^1 $$
Calculating this gives: $$ LCM = 4 \times 9 \times 5 $$
- Calculate the final result
Now, we perform the multiplication: $$ 4 \times 9 = 36 $$ $$ 36 \times 5 = 180 $$
Therefore, the least common multiple of 36 and 45 is 180.
The least common multiple (LCM) of 36 and 45 is $180$.
More Information
The least common multiple is useful when you need to find a common denominator in fractions or when solving problems that involve synchronization of different cycles. For example, if two events occur every 36 and 45 days, they will both coincide every 180 days.
Tips
- Forgetting to take the highest exponent for each prime factor.
- Mixing up the prime factorization, which can lead to incorrect calculations. To avoid these mistakes, always double-check each prime factorization and ensure you're using the highest powers correctly.
