What is the least common multiple (LCM) of 16 and 22?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 16 and 22. To find the LCM, we can use the prime factorization method or list the multiples of each number until we find the smallest common one.
Answer
The least common multiple of 16 and 22 is $176$.
Answer for screen readers
The least common multiple of 16 and 22 is $176$.
Steps to Solve
- Prime Factorization of 16
We start by finding the prime factors of 16.
The prime factorization of 16 is:
$$ 16 = 2^4 $$
- Prime Factorization of 22
Next, we find the prime factors of 22.
The prime factorization of 22 is:
$$ 22 = 2^1 \cdot 11^1 $$
- Identify the Highest Powers of Each Prime Factor
To find the LCM, we take the highest power of each prime factor found in the factorizations.
- From 16, we have $2^4$.
- From 22, we have $2^1$ and $11^1$.
The highest powers are:
- $2^4$
- $11^1$
- Calculate the LCM
The LCM is found by multiplying these highest powers together:
$$ LCM = 2^4 \cdot 11^1 $$
Calculating this gives:
$$ LCM = 16 \cdot 11 = 176 $$
The least common multiple of 16 and 22 is $176$.
More Information
The least common multiple (LCM) is the smallest number that is a multiple of both numbers. In this case, $176$ is the first number that both $16$ and $22$ can divide without a remainder.
Tips
- A common mistake is to only consider the multiples of each number and stop at the wrong value. It's essential to cover all prime factors and their highest powers to find the correct LCM.
