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Chapter 4: Problem 15

Find the LCM. 16 and 12

Short Answer

Expert verified
The LCM of 16 and 12 is 48.

Step by step solution

01

Prime Factorization

Prime factorize each number.For 16: \[ 16 = 2^4 \]For 12: \[ 12 = 2^2 \times 3^1 \]
02

Identify the Highest Powers

Determine the highest power of each prime factor that appears in the factorizations.For 2: The highest power is \(2^4\) from 16.For 3: The highest power is \(3^1\) from 12.
03

Multiply the Highest Powers

Multiply the highest powers of all prime factors identified in Step 2.Thus, the LCM is \[ 2^4 \times 3^1 = 16 \times 3 = 48 \]

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Key Concepts

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

prime factorization
To understand the Least Common Multiple (LCM) of any two numbers, the first step is 'prime factorization'.
Prime factorization means expressing a number as the product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
When we break down a number into primes, we get its prime factors.
For example:
  • The prime factorization of 16 is \( 16 = 2^4 \) because 2 is the only prime number that divides 16, and it divides 16 four times.
  • For 12, the prime factorization is \( 12 = 2^2 \times 3^1 \), because 12 can be divided by the prime number 2 twice, and by the prime number 3 once.

This step helps us understand the fundamental building blocks of each number.
highest power
The second step in finding the LCM is to identify the 'highest power' of each prime factor from the factorizations.
Each prime factor should be raised to the highest power that appears in any of the factorizations.
Let's look at our example again:
  • From the prime factorization of 16, we get \( 2^4 \).
  • From the prime factorization of 12, we get \( 2^2 \times 3^1 \).

Among the prime factor 2, the highest power is \( 2^4 \) (from 16).
Among the prime factor 3, the highest power is \( 3^1 \) (from 12).
Remember, for each prime factor, you always choose the highest power present in any number's factorization.
multiplication of factors
The final step is the 'multiplication of factors'.
This involves multiplying the highest powers of all prime factors identified.
In our example:
  • The highest power of 2 is \( 2^4 \).
  • The highest power of 3 is \( 3^1 \).

So, we multiply these together to find the LCM:
\( 2^4 \times 3^1 = 16 \times 3 = 48 \).
Therefore, the LCM of 16 and 12 is 48. This means 48 is the smallest number that is a multiple of both 16 and 12.
This method ensures that we cover all the prime factors at their highest powers, giving us the smallest common multiple.

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