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Chapter 3: Problem 11

Find the LCM of each set of numbers. $$ 30,40 $$

Short Answer

Expert verified
The LCM of 30 and 40 is 120.

Step by step solution

01

- Find Prime Factors of Each Number

First, find the prime factors of 30 and 40.- Prime factors of 30: Since 30 = 2 脳 3 脳 5- Prime factors of 40: Since 40 = 2^3 脳 5
02

- Identify the Highest Power of Each Prime

Look at all the prime factors and choose the highest power of each prime.- For 2: The highest power is 2^3.- For 3: The highest power is 3.- For 5: The highest power is 5.
03

- Calculate the LCM

Multiply the highest powers of all prime factors together to get the LCM.LCM = 2^3 脳 3 脳 5 = 8 脳 3 脳 5 = 120

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Key Concepts

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

Finding LCM
Finding the Least Common Multiple (LCM) is essential when working with multiple numbers. The LCM is the smallest number that is a multiple of each number in a given set. To find the LCM, follow these steps:
  • Identify the prime factors of each number in the set.
  • Determine the highest power of each prime factor.
  • Multiply these highest powers together.
In the exercise, the numbers given are 30 and 40. Let鈥檚 use this process to find their LCM. This method ensures you account for all the necessary multiples without missing any crucial factors.
Prime Factorization
Prime factorization is breaking down a number into its basic building blocks鈥攑rime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, to find the prime factors of 30:
  • 30 can be divided by 2 (its smallest prime factor). This gives us 2 and 15.
  • Next, divide 15 by its smallest prime factor, 3. This gives us 3 and 5.
  • Both 3 and 5 are prime numbers. So, the prime factors of 30 are 2, 3, and 5, or 2 脳 3 脳 5.
For 40, the process goes like this:
  • 40 can be divided by 2, resulting in 2 and 20.
  • 20 divided by 2 gives us 2 and 10.
  • 10 divided by 2 yields 2 and 5.
  • Since 5 is prime, the prime factors of 40 are 2, 2, 2, and 5, or 2^3 脳 5.
Understanding prime factorization helps in determining the LCM more conveniently.
Multiples
Multiples of a number are found by multiplying that number by an integer. For example, the multiples of 2 include 2, 4, 6, 8, 10, and so forth. When finding the LCM, you need to consider the multiples of each number involved. The LCM is the smallest multiple both numbers share.

Let鈥檚 consider multiples of 30 and 40:
  • Multiples of 30: 30, 60, 90, 120, ...
  • Multiples of 40: 40, 80, 120, ...

Notice that the smallest common multiple in both lists is 120. This is why identifying multiples, along with prime factors, can be so useful鈥攅specially when dealing with more complex numbers. Understanding these concepts thoroughly will simplify your work with least common multiples.

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Most popular questions from this chapter

Simplify. $$ \left(\frac{2}{3}+\frac{3}{4}\right) \div\left(\frac{5}{6}-\frac{1}{3}\right) $$

Add and simplify. $$ \frac{3}{8}+\frac{5}{12}+\frac{8}{15} $$

Simplify. Classify the given numbers as prime, composite, or neither. \([2.1 \mathrm{c}]\) $$ 1,5,7,9,14,23,43 $$

Simplify. $$ 5 \cdot 2^{2} \div 10 \quad[1.9 \mathrm{c}] $$

Subtract and simplify. $$ \frac{3}{4}-\frac{1}{8} $$
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