What is the GCF of 72 and 40?
Understand the Problem
The question is asking for the greatest common factor (GCF) of the numbers 72 and 40. The GCF is the largest number that divides both numbers without leaving a remainder. To solve it, we can find the prime factors of both numbers and determine the highest common factor among those factors.
Answer
The GCF of 72 and 40 is $8$.
Answer for screen readers
The greatest common factor (GCF) of 72 and 40 is 8.
Steps to Solve
- Find the prime factors of 72
We can start by dividing 72 by its smallest prime number, which is 2.
$$ 72 \div 2 = 36 \ 36 \div 2 = 18 \ 18 \div 2 = 9 \ 9 \div 3 = 3 \ 3 \div 3 = 1 $$
So, the prime factorization of 72 is ( 2^3 \times 3^2 ).
- Find the prime factors of 40
Next, we will do the same for 40.
$$ 40 \div 2 = 20 \ 20 \div 2 = 10 \ 10 \div 2 = 5 \ 5 \div 5 = 1 $$
So, the prime factorization of 40 is ( 2^3 \times 5^1 ).
- Identify the common factors
Now, we can list out the prime factors of both numbers:
- The prime factors of 72 are ( 2^3 ) and ( 3^2 ).
- The prime factors of 40 are ( 2^3 ) and ( 5^1 ).
The common prime factor is ( 2^3 ).
- Calculate the GCF
To find the GCF, we take the lowest power of each common prime factor.
The common prime factor is (2) and its lowest power is (3), so:
$$ GCF = 2^3 = 8 $$
The greatest common factor (GCF) of 72 and 40 is 8.
More Information
The GCF is useful in simplifying fractions and finding common denominators. Understanding how to find the GCF can be helpful in various areas of math and problem-solving scenarios.
Tips
- Forgetting to list all prime factors in the factorization steps. It's essential to ensure all prime factors are included before determining the GCF.
- Not considering the common prime factors when calculating the GCF. Always focus on the lowest power of common factors.
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