Question
Find the least common multiple. \( 3 r^{2}, 65 r^{5} \) Write your answer as a constant times a product of single variables raised to exponents.
Ask by Moreno Howell. in the United States
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The least common multiple is \( 195r^{5} \).
Solution
The least common multiple (LCM) of the two expressions \( 3r^{2} \) and \( 65r^{5} \) can be found by determining the LCM of the constants and the variables separately.
1. **LCM of the Constants:**
- The constants are 3 and 65.
- \( 65 = 5 \times 13 \)
- Since 3 is a prime number and does not divide 65, the LCM of 3 and 65 is:
\[
3 \times 5 \times 13 = 195
\]
2. **LCM of the Variables:**
- The variables are \( r^{2} \) and \( r^{5} \).
- The LCM is the variable raised to the highest power:
\[
r^{5}
\]
3. **Combine the Results:**
- Multiply the LCM of the constants by the LCM of the variables:
\[
195 \times r^{5} = 195r^{5}
\]
**Final Answer:**
\[
\boxed{195\, r^{5}}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To find the least common multiple (LCM) of the terms \(3 r^2\) and \(65 r^5\), we first identify the LCM of the coefficients (3 and 65) and then the LCM of the variable parts (\(r^2\) and \(r^5\)). The coefficients: - The LCM of 3 and 65 can be found by noticing that 65 factors into \(5 \times 13\). Since 3 and 65 have no common factors, the LCM is \(3 \times 65 = 195\). The variable parts: - For the variable \(r\), we take the variable raised to the highest power, which is \(r^5\). Putting it all together, the LCM is: \[ 195 r^5 \]
