What is the GCF of 84 and 90?

Understand the Problem

The question is asking for the greatest common factor (GCF) of the numbers 84 and 90. The GCF is the largest number that divides both numbers without leaving a remainder. We will find the prime factorization of both numbers and identify the common factors to determine the GCF.

Answer

The GCF of 84 and 90 is $6$.

Steps to Solve

Find the prime factorization of 84

To find the prime factorization of 84, we divide the number by the smallest prime number (2) until we can't divide anymore.

$$ 84 \div 2 = 42 $$ $$ 42 \div 2 = 21 $$ $$ 21 \div 3 = 7 $$ $$ 7 \div 7 = 1 $$

So, the prime factorization of 84 is $2^2 \cdot 3^1 \cdot 7^1$.

Find the prime factorization of 90

Next, we do the same for 90.

$$ 90 \div 2 = 45 $$ $$ 45 \div 3 = 15 $$ $$ 15 \div 3 = 5 $$ $$ 5 \div 5 = 1 $$

So, the prime factorization of 90 is $2^1 \cdot 3^2 \cdot 5^1$.

Identify the common prime factors

Now we compare the prime factorizations:

For 84: $2^2 \cdot 3^1 \cdot 7^1$
For 90: $2^1 \cdot 3^2 \cdot 5^1$

The common prime factors are $2$ and $3$.

Find the lowest powers of the common factors

We take the lowest power of each common prime factor:

For $2$: the minimum of $2^2$ (from 84) and $2^1$ (from 90) is $2^1$.
For $3$: the minimum of $3^1$ (from 84) and $3^2$ (from 90) is $3^1$.

Calculate the GCF

Now, we multiply these together to find the GCF:

$$ GCF = 2^1 \cdot 3^1 = 2 \cdot 3 = 6 $$

The greatest common factor (GCF) of 84 and 90 is $6$.

More Information

The GCF is useful in simplifying fractions, finding common denominators, and solving problems involving ratios.

Tips

Ignoring prime factorization: Sometimes students forget to fully factor the numbers into primes, leading to incorrect GCF. It's essential to factor fully.
Not taking the lowest power: When finding the common factors, make sure to take the lowest power of each prime factor.

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