What is the least common multiple (LCM) of 45 and 75?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 45 and 75. To solve this, we need to find the smallest multiple that both numbers share.
Answer
The LCM of 45 and 75 is 225.
Answer for screen readers
The least common multiple of 45 and 75 is 225.
Steps to Solve
- Find the prime factorization of each number
To find the LCM, we first need to determine the prime factorization of both numbers.
For 45: $$ 45 = 3^2 \times 5^1 $$
For 75: $$ 75 = 3^1 \times 5^2 $$
- Identify the highest power of each prime factor
Next, we look at each prime factor and select the highest power that appears in either factorization.
For the prime factor 3:
- Highest power: $3^2$
For the prime factor 5:
- Highest power: $5^2$
- Multiply the highest powers together
Now, we multiply these highest powers to find the LCM.
$$ \text{LCM} = 3^2 \times 5^2 $$
Calculating this gives: $$ 3^2 = 9 $$ $$ 5^2 = 25 $$
So, $$ \text{LCM} = 9 \times 25 $$
- Calculate the final result
Now we perform the multiplication: $$ 9 \times 25 = 225 $$
Thus, the least common multiple of 45 and 75 is 225.
The least common multiple of 45 and 75 is 225.
More Information
The least common multiple (LCM) is useful in various applications such as adding fractions with different denominators. In this case, 225 is significant as it represents the smallest number that both 45 and 75 can divide without leaving a remainder.
Tips
- Forgetting to take the highest power of each prime factor can result in an incorrect LCM.
- Not correctly performing the multiplication can lead to an incorrect final answer.
