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Chapter 7: Problem 77

For exercises \(55-86\), use prime factorization to find the least common multiple. $$ 120 n^{2} p^{2} ; 180 n^{5} p^{2} $$

Short Answer

Expert verified
The least common multiple is \(360 n^5 p^2\).

Step by step solution

01

Find the Prime Factorization of 120

Break down 120 into its prime factors: \[ 120 = 2^3 \times 3 \times 5 \]
02

Find the Prime Factorization of 180

Break down 180 into its prime factors: \[ 180 = 2^2 \times 3^2 \times 5 \]
03

Determine the Highest Power of Each Prime

For the least common multiple, take the highest power of each prime number appearing in the factorizations: \[ 2^3, 3^2, 5 \]
04

Consider the Variables with Their Highest Powers

In the expressions provided, the highest powers are:\[ n^5 \text{ and } p^2 \]
05

Calculate the Least Common Multiple

Combine the highest powers of all primes and variables to find the least common multiple: \[ \text{LCM} = 2^3 \times 3^2 \times 5 \times n^5 \times p^2 \]
06

Simplify the Expression

Multiply the constants to simplify the expression: \[ 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360 \] So, \[ \text{LCM} = 360 n^5 p^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factorization
Prime factorization is breaking down a number into the set of prime numbers that multiply together to give the original number.
Primes are numbers greater than 1 that have no divisors other than 1 and themselves.
Examples of prime numbers include 2, 3, 5, 7, and 11.
To perform prime factorization for 120, we would:
  • Start with the smallest prime, 2, and divide: \(120 \div 2 = 60\)
  • Continue dividing by 2: \(60 \div 2 = 30\), and again: \(30 \div 2 = 15\)
  • 15 is not divisible by 2, so we move to the next prime, 3: \(15 \div 3 = 5\)
  • Lastly, we divide by 5: \(5 \div 5 = 1\)
Each of these primes is a factor, making the prime factorization for 120: \(2^3 \times 3 \times 5\).
This breakdown helps us when finding the least common multiple (LCM) later on.
Highest Powers
When calculating the LCM, we must consider the highest powers of each prime factor found in any of the factorizations.
The highest power of a number is the greatest exponent of that prime found in the factorization steps.
In our example factorization:
  • \(2^3\) (from 120)
  • \(3^2\) (from 180)
  • \(5\) (common in both)
We use these highest powers to avoid missing any multiples.
We do the same with variables (e.g., n or p) by taking the highest exponent for each:
  • \(n^5\)\ from \(120 n^2 p^2\) and \(180 n^5 p^2\)\lags highest power is \(n^5\)
  • \(p^2\) common to both\
Using the highest powers ensures our LCM includes all necessary factors.
Variable Exponents
Variables in the expressions contribute to the LCM by taking the highest exponent of each variable.
Exponents indicate how many times a variable is multiplied by itself.
In our example:
  • 120 has exponents for n and p: \(n^2\) and \(p^2\)
  • 180 has \(n^5\) and \(p^2\)
The highest exponent of 'n' between the two is 5 (\(n^5\)), and for 'p', it is 2 (\(p^2\)).
Thus, combining these, we include \(n^5\) and \(p^2\) in the LCM.
This method ensures that the LCM is the smallest expression that holds all the necessary factors and their respective powers.
LCM Calculation
To calculate the LCM using prime factorization:
  • Find the prime factors of each number involved.
  • Identify the highest powers of each prime.
For 120 and 180, combining their factorizations:
  • 120 = \(2^3 \times 3 \times 5\)
  • 180 = \(2^2 \times 3^2 \times 5\)
The highest powers are \(2^3\), \(3^2\), and 5, as well as the highest exponents for any variables.
Then, combine:
  • Multiply the highest power of each prime: \(2^3 \times 3^2 \times 5\)
  • Include variable exponents: \(n^5 \times p^2\)
The resulting expression is: \(LCM = 2^3 \times 3^2 \times 5 \times n^5 \times p^2 = 360 n^5 p^2\).
This gives us the least common multiple, combining all necessary factors into one expression.

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Most popular questions from this chapter

For exercises \(67-82\), use the five steps and a proportion. In 2010 , about \(2,465,940\) Americans died. Find the number of Americans who died without a will. Round to the nearest hundred. (Source: www.cdc.gov, Jan. 11, 2012) Seven out of ten Americans die without a will. (Source: extension.umd.edu)

For exercises 1-10, (a) solve. (b) check. $$ \frac{4}{15} k+\frac{3}{4}=-2 $$

The relationship of the amount of salad dressing, \(x\), and the amount of sodium in the dressing, \(y\), is a direct variation. Six servings of dressing contain \(1800 \mathrm{mg}\) of sodium. a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the amount of sodium in a bottle that contains 16 servings of salad dressing. d. Use slope-intercept graphing to graph this equation. e. Use the graph to find the amount of sodium in 3 servings of salad dressing.

For exercises \(45-48\), the formula \(R=\frac{U F}{P}\) describes the glomular filtration rate by a kidney \(R\). Is the relationship of the given variables a direct variation or an inverse variation? $$ F \text { and } P \text { are constant; the relationship of } R \text { and } U \text {. } $$

For exercises \(67-82\), use the five steps and a proportion. About five of 100 pregnant women have pre-eclampsia, a condition that results in high blood pressure. About 300,000 pregnant women per year in the United States have pre-eclampsia. Find the number of pregnant women in the United States used to create this ratio. (Source: www.nytimes.com, March 17, 2009)
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