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Chapter 3: Problem 43

Find the LCM of each set of numbers. $$ 81,90 $$

Short Answer

Expert verified
The LCM of 81 and 90 is 810.

Step by step solution

01

Prime Factorization of Each Number

Begin by breaking each number into its prime factors. For 81: \[ 81 = 3^4 \] For 90: \[ 90 = 2 \times 3^2 \times 5 \]
02

Identify the Highest Powers of All Prime Factors

For each prime number that appears in the factorizations, select the highest power of that prime number. From our factorizations: - The highest power of 2 is \(2^1\) - The highest power of 3 is \(3^4\) - The highest power of 5 is \(5^1\)
03

Multiply the Highest Powers Together

Multiply the selected highest powers of all prime factors to find the LCM. \[ \text{LCM} = 2^1 \times 3^4 \times 5^1 \] Calculate it: \[ 2 \times 81 \times 5 = 2 \times 405 = 810 \]

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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 the process of expressing a number as the product of its prime factors. Prime numbers are numbers that are only divisible by 1 and themselves.

For example, let's break down the numbers in our problem:

For 81, since it can be divided by 3 continually until 1, we find:
\[ 81 = 3^4 \]

For 90, we identify the prime numbers 2, 3, and 5:
\[ 90 = 2 \times 3^2 \times 5 \]

This means 81 and 90 are expressed as products of prime factors like so:
\[ 81 = 3 \times 3 \times 3 \times 3 \] \[ 90 = 2 \times 3 \times 3 \times 5 \]
LCM Calculation
The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. It's particularly useful for solving problems involving fractions and finding common denominators.

To calculate the LCM, follow these steps:
  • List out the prime factors of each number
  • Identify the highest power of each prime factor present


From the Prime Factorization:
  • The highest power of 2 is \[2^1\]
  • The highest power of 3 is \[3^4\]
  • The highest power of 5 is \[5^1\]


Now, we multiply these highest powers together to get the LCM:
\[ \text{LCM} = 2^1 \times 3^4 \times 5^1 = 2 \times 81 \times 5 = 2 \times 405 = 810 \]
Mathematics Steps
Students often find the step-by-step process helpful in solving mathematical problems. Let's outline the steps once more clearly:

Step 1: Perform Prime Factorization
  • 81 = \[3^4\]
  • 90 = \[2 \times 3^2 \times 5\]

Step 2: Identify the Highest Powers:
  • Highest power of 2 is \[2^1\]
  • Highest power of 3 is \[3^4\]
  • Highest power of 5 is \[5^1\]

Step 3: Multiply the Highest Powers:
  • LCM = \[2^1 \times 3^4 \times 5^1 = 810\]

Breaking down the process into manageable steps makes it easier to understand. Additionally, using prime factorization helps in simplifying and solving many mathematical problems related to multiples.

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Most popular questions from this chapter

Using 45-in. fabric, Regan needs \(1 \frac{3}{8}\) yd for a dress, \(\frac{5}{8}\) yd of contrasting fabric for the band at the bottom, and \(3 \frac{3}{8}\) yd for a coordinating jacket. How many yards of 45 -in. fabric are needed in all?

Subtract and simplify. $$ \frac{3}{4}-\frac{3}{20} $$

Temperature. Fahrenheit temperature can be obtained from Celsius (Centigrade) temperature by multiplying by \(1 \frac{4}{5}\) and adding \(32^{\circ} .\) What Fahrenheit temperature corresponds to the Celsius temperature of boiling water, \(100^{\circ} ?\)

Estimate each of the following by estimating each mixed numeral as a whole number or as a mixed numeral where the fraction part is \(\frac{1}{2}\) and by estimating each fraction as \(0, \frac{1}{2},\) or 1 . $$ 16 \frac{1}{5} \div 2 \frac{1}{11}+25 \frac{9}{10}-4 \frac{11}{23} $$

Applications of LCMs: Planet Orbits. Jupiter, Saturn, and Uranus all revolve around the sun. Jupiter takes 12 years, Saturn 30 years, and Uranus 84 years to make a complete revolution. On a certain night, you look at Jupiter, Saturn, and Uranus and wonder how many years it will take before they have the same positions again. How often will Jupiter, Saturn, and Uranus appear in the same direction in the night sky as seen from the earth?
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