What is the greatest common factor of 72 and 40?
Understand the Problem
The question is asking for the greatest common factor (GCF), which means we need to find the largest number that divides both 72 and 40 without leaving a remainder. We will approach this by determining the prime factorization of both numbers and identifying the common 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
- Prime Factorization of 72
First, let's find the prime factors of 72. We can do this by dividing by the smallest prime numbers.
$$ 72 = 2 \times 36 $$
$$ 36 = 2 \times 18 $$
$$ 18 = 2 \times 9 $$
$$ 9 = 3 \times 3 $$
So, the prime factorization of 72 is:
$$ 72 = 2^3 \times 3^2 $$
- Prime Factorization of 40
Next, we find the prime factors of 40 in the same way.
$$ 40 = 2 \times 20 $$
$$ 20 = 2 \times 10 $$
$$ 10 = 2 \times 5 $$
So, the prime factorization of 40 is:
$$ 40 = 2^3 \times 5^1 $$
- Identify Common Factors
Now we identify the common prime factors in both factorizations.
- For (2): The minimum power is (2^3).
- For (3): It does not appear in the factorization of 40.
- For (5): It does not appear in the factorization of 72.
Thus, the only common prime factor is (2^3).
- Calculate the GCF
Now we calculate the greatest common factor:
$$ \text{GCF} = 2^3 = 8 $$
The greatest common factor (GCF) of 72 and 40 is $8$.
More Information
The greatest common factor is useful in simplifying fractions and finding common denominators. It's an important concept in various areas of math, including number theory and algebra. Notably, the GCF can also be utilized in real-life scenarios, such as determining how to evenly distribute items among groups.
Tips
- Forgetting to include all prime factors when calculating the GCF.
- Confusing the GCF with the least common multiple (LCM). Remember, GCF looks for common factors while LCM looks for the smallest multiple.