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Chapter 5: Problem 108

The process of finding the greatest common divisor of two natural numbers is similar to finding the least common multiple of the numbers. Describe how the two processes differ.

Short Answer

Expert verified
The processes differ in how the prime factors are utilised. For the GCD, only the common prime factors are used, while for the LCM, all the prime factors are used, each as many times as it appears in either breakdown.

Step by step solution

01

Understanding the GCD and LCM

The GCD of two natural numbers is the largest number that can divide both numbers without leaving a remainder. On the other hand, the LCM of two natural numbers is the smallest number which can be divided by both numbers without leaving a remainder.
02

Finding the GCD

For two numbers, we list their prime factors. The GCD is found by multiplying all the common prime factors. For example, the GCD of 18 (which is \(2 \times 3^2\)) and 24 (which is \(2^3 \times 3\)) is \(2 \times 3 = 6\), because 2 and 3 are common prime factors.
03

Finding the LCM

Again, list the prime factors of the two numbers. The LCM is found by multiplying all the prime factors of both numbers, but each factor is used the greatest number of times it appears in either breakdown. Like for numbers 18 and 24, 2 appears three times, whilst 3 appears twice. So the LCM is \(2^3 \times 3^2 = 72\).
04

Comparing the Processes

Both processes involve finding the prime factors of the numbers. However, for the GCD, we multiply only the common factors and for the LCM, we multiply all the prime factors, each as many times as it appears in either breakdown.

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Most popular questions from this chapter

The bar graph shows changes in the percentage of college graduates for Americans ages 25 and older from 1990 to 2010. Exercises 125-126 involve developing arithmetic sequences that model the data. In \(1990,24.4 \%\) of American men ages 25 and older had graduated from college. On average, this percentage has increased by approximately \(0.3\) each year. a. Write a formula for the \(n\)th term of the arithmetic sequence that models the percentage of American men ages 25 and older who had graduated from college \(n\) years after \(1989 .\) b. Use the model from part (a) to project the percentage of American men ages 25 and older who will be college graduates by \(2019 .\)

Find the indicated term for the geometric sequence with first term, \(a_{1}\), and common ratio, \(r\). Find \(a_{200}\), when \(a_{1}=60, r=1\).

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(3,12,48,192, \ldots\)

Write a formula for the general term (the nth term) of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7}\), the seventh term of the sequence. \(1.5,-3,6,-12, \ldots\)

The sum, \(S_{n}\), of the first n terms of an arithmetic sequence is given by $$ S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right), $$ in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by $$ S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r}, $$ in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(4,10,16,22, \ldots\)
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