What is the GCF 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 positive integer that divides both numbers without leaving a remainder.
Answer
The GCF of 45 and 81 is $9$.
Answer for screen readers
The greatest common factor (GCF) of 45 and 81 is $9$.
Steps to Solve
- Find the prime factorization of each number
To find the GCF, we start by determining the prime factors of both numbers:
For 45:
- It can be factored as $45 = 3 \times 15 = 3 \times 3 \times 5 = 3^2 \times 5$.
For 81:
- It can be factored as $81 = 9 \times 9 = 3 \times 3 \times 3 \times 3 = 3^4$.
Thus, the prime factorizations are:
- $45 = 3^2 \times 5$
- $81 = 3^4$
- Identify the common prime factors
Next, we look for the prime factors that are common to both 45 and 81. The only common factor in their factorizations is $3$.
- Determine the lowest powers of common prime factors
To find the GCF, we take the lowest power of the common prime factor:
- The power of 3 in 45 is $2$ (from $3^2$) and in 81 is $4$ (from $3^4$). Thus, the lowest power is $3^2$.
- Calculate the GCF
Now, we calculate the GCF using the lowest power: $$ GCF = 3^2 = 9 $$
The greatest common factor (GCF) of 45 and 81 is $9$.
More Information
The greatest common factor helps in simplifying fractions and finding common denominators. It's particularly useful in fraction reduction.
Tips
- Confusing GCF with least common multiple (LCM). The GCF is the largest number that divides both, while LCM is the smallest number that both can divide into.
- Forgetting to consider all prime factors, especially when numbers have different primes.
