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Chapter 7: Problem 39

Find the LCM. $$8,9$$

Short Answer

Expert verified
The LCM of 8 and 9 is 72.

Step by step solution

01

Prime Factorization

Prime factorize each number to find the smallest positive integer divisible by both 8 and 9.\(8 = 2^3\)\(9 = 3^2\)
02

Identify All Prime Factors

List all the different prime factors involved. Here, the prime factors are 2 and 3.
03

Determine the Highest Powers

For each prime factor, determine the highest power that appears in the factorization of each number.\(2^3\) is the highest power of 2,\(3^2\) is the highest power of 3.
04

Multiply the Highest Powers

Multiply these highest powers together to find the LCM.\[\text{LCM} = 2^3 \times 3^2 = 8 \times 9 = 72\]

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Key Concepts

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

Prime Factorization
Prime factorization breaks a number down into the set of prime numbers that, when multiplied together, give the original number. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, to prime factorize 8, we find it consists of three 2s: \(8 = 2^3\). Likewise, the prime factorization of 9 is: \(9 = 3^2\). Prime factorization helps identify the building blocks of numbers and is a necessary step in finding the Least Common Multiple (LCM) of two or more numbers.
Highest Powers
Once we have the prime factors of each number, the next step is to determine the highest powers of these factors. The highest power is simply the greatest exponent listed for each prime number in the factorizations.

For our example:
  • The prime factor 2 appears in 8 with a power of 3 (written as \(2^3\)).
  • The prime factor 3 appears in 9 with a power of 2 (written as \(3^2\)).
To find the LCM, we need to take the highest power of each prime that appears in any of the factorizations.
Multiplication of Primes
After determining the highest powers of all the prime factors, finding the LCM involves multiplying these highest powers together. This product gives us the smallest number that is evenly divisible by all original numbers.

For our example, we multiply the highest powers of 2 and 3:
  • The highest power of 2 is \(2^3\).
  • The highest power of 3 is \(3^2\).
Therefore, the LCM is calculated as:

\(\text{LCM} = 2^3 \times 3^2 = 8 \times 9 = 72\)
This result means that 72 is the smallest number that can be evenly divided by both 8 and 9.

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