How to find the greatest common factor with exponents?
Understand the Problem
The question is asking how to determine the greatest common factor (GCF) of two or more numbers that include exponents. This typically involves finding the prime factorization of each number and identifying the lowest powers of the common prime factors.
Answer
The GCF of 18 and 24 is \( 6 \).
Answer for screen readers
The greatest common factor (GCF) of 18 and 24 is ( 6 ).
Steps to Solve
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Find the Prime Factorization of Each Number
First, we need to express each number as a product of its prime factors. For example, for the numbers 18 and 24, we have:- ( 18 = 2^1 \times 3^2 )
- ( 24 = 2^3 \times 3^1 )
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Identify Common Prime Factors
Next, look for the prime factors that are common to both numbers. From the factorizations above, we can see that the common prime factors are ( 2 ) and ( 3 ). -
Take the Lowest Power of Each Common Prime Factor
For each common prime factor, take the minimum exponent from the factorizations:- For ( 2 ): the lowest power is ( 2^1 ) (from 18)
- For ( 3 ): the lowest power is ( 3^1 ) (from 24)
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Multiply the Results
Finally, multiply the results of the lowest powers of the common prime factors to get the GCF:
$$ \text{GCF} = 2^1 \times 3^1 = 2 \times 3 = 6 $$
The greatest common factor (GCF) of 18 and 24 is ( 6 ).
More Information
The GCF is useful in various applications, such as simplifying fractions and finding common denominators. Notably, the GCF of two numbers is the largest number that can divide both numbers without leaving a remainder.
Tips
- Forgetting to consider the exponents when identifying the lowest power of common factors.
- Miscalculating the prime factorization of the numbers.
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