What is the greatest common factor of 45 and 81?
Understand the Problem
The question is asking for the greatest common factor (GCF) of the numbers 45 and 81, which involves finding the largest number that divides both of them without leaving a remainder.
Answer
$9$
Answer for screen readers
The greatest common factor (GCF) of 45 and 81 is $9$.
Steps to Solve
- Prime Factorization of 45
First, we'll find the prime factorization of 45.
The factors of 45 can be found by dividing by the smallest prime number: $$ 45 \div 3 = 15 $$ $$ 15 \div 3 = 5 $$ 5 is a prime number.
So the prime factorization of 45 is: $$ 45 = 3^2 \cdot 5^1 $$
- Prime Factorization of 81
Next, we find the prime factorization of 81.
Again, start with the smallest prime number: $$ 81 \div 3 = 27 $$ $$ 27 \div 3 = 9 $$ $$ 9 \div 3 = 3 $$ $$ 3 \div 3 = 1 $$
So the prime factorization of 81 is: $$ 81 = 3^4 $$
- Identify Common Factors
Now, let's identify the common prime factors in the prime factorizations of both numbers.
From earlier:
- The prime factorization of 45 is: $3^2 \cdot 5^1$
- The prime factorization of 81 is: $3^4$
The common prime factor is $3$.
- Determine the Greatest Common Factor
To find the GCF, we take the lowest power of the common prime factor: For $3$, the minimum exponent is $2$.
Thus, the GCF is: $$ GCF = 3^2 = 9 $$
The greatest common factor (GCF) of 45 and 81 is $9$.
More Information
The GCF helps in simplifying fractions, finding common denominators, and is a fundamental concept in number theory. The prime factorization method ensures accurate identification of common factors.
Tips
- Not Using Prime Factorization: Some may attempt to find the GCF by listing the factors of each number without using prime factorization, which can be inefficient.
- Ignoring Multiple Factors: It’s crucial to check that all common factors are considered, particularly when they have different exponents.
