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Chapter 3: Problem 36

Find the LCM of each set of numbers. $$ 12,72 $$

Short Answer

Expert verified
The LCM of 12 and 72 is 72.

Step by step solution

01

Find the Prime Factorization

Break down each number into its prime factors. For 12: 1. 12 梅 2 = 6 2. 6 梅 2 = 3 3. 3 is a prime number, so we stop here. Hence, the prime factorization of 12 is 2^2 * 3.For 72: 1. 72 梅 2 = 36 2. 36 梅 2 = 18 3. 18 梅 2 = 9 4. 9 梅 3 = 3 5. 3 is a prime number, so we stop here. Thus, the prime factorization of 72 is 2^3 * 3^2.
02

Identify the Greatest Power of Each Prime Factor

For each prime number that appears in the factorizations, take the highest power of that prime. From 12: 2^2, 3^1 From 72: 2^3, 3^2 Therefore, the highest power of each prime factor is 2^3 and 3^2.
03

Calculate the LCM

Multiply the highest powers of all prime factors to get the LCM. LCM = 2^3 * 3^2 2^3 = 8 3^2 = 9 Therefore, LCM = 8 * 9 = 72.

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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 breaking down a number into its basic building blocks, which are prime numbers. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. For example, let's take the number 12. To perform its prime factorization, we start by dividing 12 by the smallest prime number, which is 2.
  • 12 梅 2 = 6
  • 6 梅 2 = 3
  • 3 is a prime number
So, the prime factorization of 12 is 2^2 * 3.
Now let's do the same for 72:
  • 72 梅 2 = 36
  • 36 梅 2 = 18
  • 18 梅 2 = 9
  • 9 梅 3 = 3
  • 3 is a prime number
Thus, the prime factorization of 72 is 2^3 * 3^2.
Using prime factorization simplifies finding common denominators, least common multiples, and greatest common divisors.
Greatest Power
The next step after prime factorization is to identify the greatest power of each prime factor. This helps us find the Least Common Multiple (LCM). For each different prime factor that appears in any of the numbers, we select the highest power that appears.
  • From 12: we have 2^2 and 3^1
  • From 72: we have 2^3 and 3^2
The greatest powers are those with the highest exponents:
  • The greatest power for the prime number 2 is 2^3
  • The greatest power for the prime number 3 is 3^2
Combining these, the highest powers of all prime factors are 2^3 and 3^2.
Educational Math Problem
To tie it all together, let's solve an educational math problem by computing the Least Common Multiple (LCM). The LCM is the smallest number that both original numbers can divide without leaving a remainder.
Using our previous steps of prime factorization and finding the greatest power, we multiply the highest powers of each prime factor:
  • LCM = 2^3 * 3^2
  • 2^3 = 8
  • 3^2 = 9
  • So, LCM = 8 * 9 = 72

The Least Common Multiple of 12 and 72 is thus 72. Understanding how to find the LCM helps in solving problems involving fractions, least common denominators, and more. Once you grasp these fundamental techniques, many math problems become easier to handle.

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Most popular questions from this chapter

Add and simplify. $$ \frac{1}{2}+\frac{3}{8}+\frac{1}{4} $$

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

Add and simplify. $$ \frac{1}{10}+\frac{2}{100}+\frac{3}{1000} $$

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 . $$ 1 \frac{5}{8}+1 \frac{27}{28} \cdot 6 \frac{35}{74} $$

Consider only the numbers \(3,4,5,\) and 6 each can be placed in a blank in the following. $$ \square+\frac{\square}{\square} \cdot \square=? $$ What placement of the numbers in the blanks yields the largest sum?
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