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Chapter 3: Problem 22

Find the LCM of each set of numbers. $$ 3,5,7 $$

Short Answer

Expert verified
The LCM of 3, 5, and 7 is 105.

Step by step solution

01

Understand the Definition of LCM

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers.
02

List the Multiples of Each Number

List the first few multiples of each number. Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, ...Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, ...Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, ...
03

Identify the Least Common Multiple

Find the smallest number that appears in all of the lists of multiples from Step 2. The least common multiple of 3, 5, and 7 is 105, since 105 is the smallest number that is a multiple of 3, 5, and 7.

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Key Concepts

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

LCM calculation
The Least Common Multiple (LCM) is a fundamental concept in mathematics, particularly useful for solving problems involving fractions, ratios, and multiple measurements. It is the smallest number that is evenly divisible by each member of a set of numbers. For instance, to find the LCM of 3, 5, and 7, we must first understand its calculation process.
First, list the multiples of each number:
  • Multiples of 3: 3, 6, 9, 12, 15, 18, 21, ...
  • Multiples of 5: 5, 10, 15, 20, 25, 30, 35, ...
  • Multiples of 7: 7, 14, 21, 28, 35, 42, 49, ...

The smallest number present in all lists is the Least Common Multiple. Here, it is found by checking the common multiples from the lists. The answer is 105. Understanding this step-by-step method strengthens the foundation and makes LCM problems easier to solve.
multiples
Multiples are numbers obtained by multiplying a given number by the set of natural numbers. They form the base for finding the LCM. Let's delve deeper to comprehend this concept better.
For example, taking the number 3, its multiples can be obtained by multiplying it with natural numbers:
  • 3 脳 1 = 3
  • 3 脳 2 = 6
  • 3 脳 3 = 9
  • ...

Similarly, the multiples of 5 and 7 are:
  • 5 脳 1 = 5
  • 5 脳 2 = 10
  • 5 脳 3 = 15
  • ...
  • 7 脳 1 = 7
  • 7 脳 2 = 14
  • 7 脳 3 = 21
  • ...

By listing these multiples, we set the stage to determine the Least Common Multiple. Multiples frequently appear in different math problems, so understanding them is crucial.
common multiples
Common multiples are multiples that two or more numbers share. Identifying these is integral to locating the Least Common Multiple.
To illustrate, let鈥檚 revisit the multiples of 3, 5, and 7:

  • Multiples of 3: 3, 6, 9, 12, 15, ...
  • Multiples of 5: 5, 10, 15, 20, ...
  • Multiples of 7: 7, 14, 21, 28, ...

From these lists, identify the first common multiples. By comparing them, we find:
  • 3 and 5 share multiple 15
  • 3 and 7 share multiple 21
  • 5 and 7 share multiple 35

However, since we need the least number common to all: we continue comparison till we find 105 appearing as common across all. Hence, 105 is the LCM.
Understanding common multiples helps in diverse mathematical contexts, such as addition or subtraction of fractions, alignment of periodic events, and problem-solving requiring synchronization.

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Most popular questions from this chapter

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

Simplify. $$ \frac{1}{2}-\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{2}\right)^{3} $$

Vertical Leaps. Eight-year-old Zachary registered vertical leaps of \(12 \frac{3}{4}\) in. \(13 \frac{3}{4}\) in., \(13 \frac{1}{2}\) in., and 14 in. Find ais average vertical leap.

Add and simplify. $$ \frac{3}{16}+\frac{5}{16}+\frac{4}{16} $$

Add and simplify. $$ \frac{3}{8}+\frac{1}{8}+\frac{2}{8} $$
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