What is the GCF of 18 and 60?
Understand the Problem
The question is asking for the greatest common factor (GCF) of the numbers 18 and 60. To find the GCF, we will identify the largest number that can divide both 18 and 60 without leaving a remainder.
Answer
The GCF of 18 and 60 is $6$.
Answer for screen readers
The greatest common factor (GCF) of 18 and 60 is $6$.
Steps to Solve
- Find the prime factorization of 18
To begin, we need to find the prime factors of 18. We can do this by dividing 18 by the smallest prime number until we reach 1.
$$ 18 \div 2 = 9 $$ $$ 9 \div 3 = 3 $$ $$ 3 \div 3 = 1 $$
So, the prime factorization of 18 is: $$ 18 = 2^1 \cdot 3^2 $$
- Find the prime factorization of 60
Next, we will find the prime factors of 60 in the same way.
$$ 60 \div 2 = 30 $$ $$ 30 \div 2 = 15 $$ $$ 15 \div 3 = 5 $$ $$ 5 \div 5 = 1 $$
So, the prime factorization of 60 is: $$ 60 = 2^2 \cdot 3^1 \cdot 5^1 $$
- Identify the common prime factors
Now, we need to identify the common prime factors between the two numbers.
- For 2: The lowest exponent is $1$ (from 18).
- For 3: The lowest exponent is $1$ (from 60).
- The factor 5 is not common to both.
Thus, the common prime factors are: $$ 2^1 \cdot 3^1 $$
- Calculate the GCF
Now we can calculate the GCF by multiplying the common prime factors together:
$$ GCF = 2^1 \cdot 3^1 = 2 \cdot 3 = 6 $$
The greatest common factor (GCF) of 18 and 60 is $6$.
More Information
The GCF is important in various applications, such as simplifying fractions or finding common denominators. Understanding GCF helps in creating basic mathematical operations and is foundational in number theory.
Tips
- Failing to find the correct prime factorization for each number can lead to an incorrect GCF.
- Forgetting to consider the lowest exponent when identifying common factors can also result in errors.