Find the greatest common factor and least common multiple of A, B, and C. Write your answers as a product of powers of primes in increasing order. A = 26-57-175 B=36-7⁰-11³-13-197 C=54-75-137-176-195 GCF(A,B,C) = LCM(A,B,C) =
Find the greatest common factor and least common multiple of A, B, and C. Write your answers as a product of powers of primes in increasing order. A = 26-57-175 B=36-7⁰-11³-13-197 C=54-75-137-176-195 GCF(A,B,C) = LCM(A,B,C) =
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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Question
![**Finding the Greatest Common Factor (GCF) and Least Common Multiple (LCM)**
**Problem Statement:**
Find the greatest common factor and least common multiple of \(A\), \(B\), and \(C\). Write your answers as a product of powers in increasing order.
**Given:**
\[
A = 2^6 \cdot 5^7 \cdot 17^5
\]
\[
B = 3^6 \cdot 7^9 \cdot 11^3 \cdot 13 \cdot 19^7
\]
\[
C = 5^4 \cdot 7^5 \cdot 13^7 \cdot 17^6 \cdot 19^5
\]
**Task:**
Calculate:
\[
\text{GCF}(A,B,C) =
\]
\[
\text{LCM}(A,B,C) =
\]
[Text input fields for answers]
**Explanation for Students:**
***Greatest Common Factor (GCF):***
The GCF is the highest factor that divides all given numbers without leaving a remainder. In terms of prime factors, it is found by taking the lowest powers of all primes common to the given numbers.
***Least Common Multiple (LCM):***
The LCM is the smallest multiple shared by the given numbers. To find the LCM using prime factorization, you take the highest power of each prime that appears in the factorization of any of the given numbers.
***Instructions to Solve:***
1. **Identify common primes** in all three numbers and compare their powers.
2. For the **GCF**, select the smallest exponent for each common prime.
3. For the **LCM**, select the largest exponent for each prime.
**Example Calculation:**
- For GCF:
- Compare \(5\) in \(A\), \(B\), and \(C\): \(5^7\), \(5^0\) (since not present in B), and \(5^4\) -> smallest is \(5^0\)
- Similarly, compare powers for other primes.
- For LCM:
- Compare \(5\) in \(A\), \(B\), and \(C\): \(5^7\), \(5^0\), and \(5^4\) -> largest is \(5^](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6ee3c1cf-5502-4aa3-81a2-22fca9c39339%2F30ba149a-dc92-439e-939c-84593657452b%2F9z1nqzd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Greatest Common Factor (GCF) and Least Common Multiple (LCM)**
**Problem Statement:**
Find the greatest common factor and least common multiple of \(A\), \(B\), and \(C\). Write your answers as a product of powers in increasing order.
**Given:**
\[
A = 2^6 \cdot 5^7 \cdot 17^5
\]
\[
B = 3^6 \cdot 7^9 \cdot 11^3 \cdot 13 \cdot 19^7
\]
\[
C = 5^4 \cdot 7^5 \cdot 13^7 \cdot 17^6 \cdot 19^5
\]
**Task:**
Calculate:
\[
\text{GCF}(A,B,C) =
\]
\[
\text{LCM}(A,B,C) =
\]
[Text input fields for answers]
**Explanation for Students:**
***Greatest Common Factor (GCF):***
The GCF is the highest factor that divides all given numbers without leaving a remainder. In terms of prime factors, it is found by taking the lowest powers of all primes common to the given numbers.
***Least Common Multiple (LCM):***
The LCM is the smallest multiple shared by the given numbers. To find the LCM using prime factorization, you take the highest power of each prime that appears in the factorization of any of the given numbers.
***Instructions to Solve:***
1. **Identify common primes** in all three numbers and compare their powers.
2. For the **GCF**, select the smallest exponent for each common prime.
3. For the **LCM**, select the largest exponent for each prime.
**Example Calculation:**
- For GCF:
- Compare \(5\) in \(A\), \(B\), and \(C\): \(5^7\), \(5^0\) (since not present in B), and \(5^4\) -> smallest is \(5^0\)
- Similarly, compare powers for other primes.
- For LCM:
- Compare \(5\) in \(A\), \(B\), and \(C\): \(5^7\), \(5^0\), and \(5^4\) -> largest is \(5^
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