What is the greatest common factor of 84 and 90?
Understand the Problem
The question is asking to find the greatest common factor (GCF) of the numbers 84 and 90. The GCF is the highest number that can exactly divide both numbers without any remainder. We will find the prime factors of each number and determine the common factors to identify the GCF.
Answer
6
Answer for screen readers
The greatest common factor (GCF) of 84 and 90 is 6.
Steps to Solve
- Find the prime factors of 84
To find the prime factors of 84, we can divide it by the smallest prime numbers until we reach 1.
$$ 84 \div 2 = 42 $$
$$ 42 \div 2 = 21 $$
$$ 21 \div 3 = 7 $$
$$ 7 \div 7 = 1 $$
Thus, the prime factorization of 84 is $2^2 \times 3^1 \times 7^1$.
- Find the prime factors of 90
Next, we will find the prime factors of 90 in a similar way.
$$ 90 \div 2 = 45 $$
$$ 45 \div 3 = 15 $$
$$ 15 \div 3 = 5 $$
$$ 5 \div 5 = 1 $$
Thus, the prime factorization of 90 is $2^1 \times 3^2 \times 5^1$.
- Identify the common prime factors
Now we compare the prime factorizations of both numbers:
- For 84: $2^2 \times 3^1 \times 7^1$
- For 90: $2^1 \times 3^2 \times 5^1$
The common prime factors are:
- $2^{\text{min}(2,1)} = 2^1$
- $3^{\text{min}(1,2)} = 3^1$
- Calculate the GCF
Now, we multiply the lowest powers of the common prime factors:
$$ GCF = 2^1 \times 3^1 = 2 \times 3 = 6 $$
Thus, the greatest common factor of 84 and 90 is 6.
The greatest common factor (GCF) of 84 and 90 is 6.
More Information
The GCF is useful in various applications, such as simplifying fractions and finding common denominators. Understanding how to derive the GCF can help with a profound knowledge of number theory and its applications in algebra.
Tips
- Forgetting to consider all prime factors of both numbers.
- Miscalculating the minimum exponent for common factors when they are compared.