Build the least common multiple of A, B, and C using the example/method in module 8 on page 59&60. Then write the prime factorization of the least common multiple of A, B, and C. A=2-3-7213³ B = 38132199 - 237 C = 5-11-17³-23²
Build the least common multiple of A, B, and C using the example/method in module 8 on page 59&60. Then write the prime factorization of the least common multiple of A, B, and C. A=2-3-7213³ B = 38132199 - 237 C = 5-11-17³-23²
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Building the Least Common Multiple (LCM) Using Prime Factorization**
To find the least common multiple (LCM) of three given numbers, A, B, and C, we will use their prime factorizations.
Given:
- \( A = 2 \cdot 3 \cdot 7^2 \cdot 13^3 \)
- \( B = 3^8 \cdot 13^2 \cdot 19^9 \cdot 23^7 \)
- \( C = 5 \cdot 11 \cdot 17^3 \cdot 23^2 \)
### Steps to find the LCM:
1. **Identify all the distinct prime factors** in the factorizations of A, B, and C.
2. **For each distinct prime factor, select the highest power of that prime** that appears in any of the factorizations.
### Prime Factors and their Highest Powers:
- **2**: Only appears in A, so the highest power is \(2^1\).
- **3**: Appears in A and B. The highest power is \(3^8\).
- **7**: Appears only in A, so the highest power is \(7^2\).
- **13**: Appears in A and B. The highest power is \(13^3\).
- **19**: Only appears in B, so the highest power is \(19^9\).
- **23**: Appears in B and C. The highest power is \(23^7\).
- **5**: Only appears in C, so the highest power is \(5^1\).
- **11**: Only appears in C, so the highest power is \(11^1\).
- **17**: Only appears in C, so the highest power is \(17^3\).
### Write the Prime Factorization of the LCM:
\[ \text{LCM} = 2^1 \cdot 3^8 \cdot 7^2 \cdot 13^3 \cdot 19^9 \cdot 23^7 \cdot 5^1 \cdot 11^1 \cdot 17^3 \]
This is the prime factorization of the least common multiple of A, B, and C.
### Diagram or Graph:
There are no](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6ee3c1cf-5502-4aa3-81a2-22fca9c39339%2F3ca50f6e-a582-4f10-a8fb-c1637b451817%2Fcyxi375_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Building the Least Common Multiple (LCM) Using Prime Factorization**
To find the least common multiple (LCM) of three given numbers, A, B, and C, we will use their prime factorizations.
Given:
- \( A = 2 \cdot 3 \cdot 7^2 \cdot 13^3 \)
- \( B = 3^8 \cdot 13^2 \cdot 19^9 \cdot 23^7 \)
- \( C = 5 \cdot 11 \cdot 17^3 \cdot 23^2 \)
### Steps to find the LCM:
1. **Identify all the distinct prime factors** in the factorizations of A, B, and C.
2. **For each distinct prime factor, select the highest power of that prime** that appears in any of the factorizations.
### Prime Factors and their Highest Powers:
- **2**: Only appears in A, so the highest power is \(2^1\).
- **3**: Appears in A and B. The highest power is \(3^8\).
- **7**: Appears only in A, so the highest power is \(7^2\).
- **13**: Appears in A and B. The highest power is \(13^3\).
- **19**: Only appears in B, so the highest power is \(19^9\).
- **23**: Appears in B and C. The highest power is \(23^7\).
- **5**: Only appears in C, so the highest power is \(5^1\).
- **11**: Only appears in C, so the highest power is \(11^1\).
- **17**: Only appears in C, so the highest power is \(17^3\).
### Write the Prime Factorization of the LCM:
\[ \text{LCM} = 2^1 \cdot 3^8 \cdot 7^2 \cdot 13^3 \cdot 19^9 \cdot 23^7 \cdot 5^1 \cdot 11^1 \cdot 17^3 \]
This is the prime factorization of the least common multiple of A, B, and C.
### Diagram or Graph:
There are no
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