LCM of 22 and 16
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 22 and 16. To solve this, we will find the multiples of both numbers and determine the smallest multiple that they have in common.
Answer
The least common multiple (LCM) of 22 and 16 is 176.
Answer for screen readers
The least common multiple (LCM) of 22 and 16 is 176.
Steps to Solve
- Find the prime factorization of each number
To begin, we factor both numbers into their prime components.
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For 22, the prime factorization is: $$ 22 = 2 \times 11 $$
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For 16, the prime factorization is: $$ 16 = 2^4 $$
- Identify the highest power of each prime factor
Next, we identify the highest power of each prime factor that appears in the factorizations of both numbers.
- The prime factors we have are 2 and 11.
- The highest power of 2 is $2^4$ (from 16).
- The highest power of 11 is $11^1$ (from 22).
- Calculate the LCM
Now we can calculate the least common multiple by multiplying the highest powers of all prime factors together:
$$ \text{LCM} = 2^4 \times 11^1 $$
- Evaluate the expression
Finally, we evaluate the expression to find the actual LCM:
$$ \text{LCM} = 16 \times 11 = 176 $$
The least common multiple (LCM) of 22 and 16 is 176.
More Information
The least common multiple is useful in various applications, such as finding common denominators in fractions or solving problems involving repeated events. The LCM of two numbers can also help in simplifying ratios.
Tips
- A common mistake is to list multiples of the numbers and find the smallest common value without using prime factorization. This may lead to errors, especially with larger numbers.
- Another mistake is to miscalculate prime factorizations, leading to an incorrect LCM.
