Find the factors of 72 and 80; the greatest common factor is the greatest number in both lists.

$\displaystyle 72: 1,2,3,4,6,\underline{8},9,12,18,24,36,72$

$\displaystyle 80:1,2,4,5,\underline{8},10,16,20,40,80$

$\displaystyle \textrm{GCF}(72,80) = 8$

Fully factor $\displaystyle 288$ and $\displaystyle 528$. Circle the factors in common:

Multiple the common factors to find the greatest common factor:

$\displaystyle GCF=2\times2\times2\times2\times3=48$

We are looking for a number that shares no factors with 10 other than 1. 20 and 28 can be eliminated since 2 divides 10, 20, 28. 25 and 35 can be eliminated since 5 divides 10, 25, and 25.

21 is the correct choice; $\displaystyle 10 = 2\times 5$ and $\displaystyle 21 = 3 \times 7$. Since they share no prime factors, $\displaystyle GCF (10,21) = 1$, and 10 and 21 are relatively prime.

For a number $\displaystyle N$ to be relatively prime with $\displaystyle 35$, $\displaystyle GCF (N,35) = 1$. Equivalently, $\displaystyle N$ and $\displaystyle 35$ cannot share a prime factor. $\displaystyle 35 = 5 \times 7$, so any number that does not have either $\displaystyle 5$ or $\displaystyle 7$ as a factor is a correct choice.

We can eliminate $\displaystyle 30$ and $\displaystyle 40$, since $\displaystyle 5$ is a factor of each; we can eliminate $\displaystyle 28$ and $\displaystyle 42$, since $\displaystyle 7$ is a factor of each. But $\displaystyle 27 = 3 \times 3\times 3$, so $\displaystyle GCF (27,35) = 1$. $\displaystyle 27$ is the correct choice.

To find the greatest common factor (GCF), you need to determine the factor that both numbers share that is of the greatest value. List the factors of each number and identify the largest number in value that is in both lists:

$\displaystyle 20: 1, 2, 4, 5, 10, 20$

$\displaystyle 36: 1, 2, 3, 4, 6, 9, 12, 18, 36$

The GCF of 20 and 36 is 4 since it is the largest number in value that shows up in both lists.

To find the greatest common factor (GCF), you need to determine the factor that all three numbers share that is of the greatest value. List the factors of each number and identify the largest number in value that is in all three lists:

$\displaystyle 15: 1, 3, 5, 15$

$\displaystyle 30: 1, 2, 3, 5, 6, 10, 15, 30$

$\displaystyle 40: 1, 2, 4, 5, 8, 10, 20, 40$

The GCF of 15, 30, and 40 is 5 since it is the largest number in value that shows up in all three lists.

Factor each number:

Identify the common factors ($\displaystyle 2,2,3$). The greatest common factor is the product of all of the common factors.

$\displaystyle GCF=2\times2\times3=12$

The greatest common factor of two numbers is the largest shared factor that the two numbers have in common. 

The factors of 36 are $\displaystyle 1, 36; 2, 18; 3, 12; 4, 9; 6$

The factors of 108 are $\displaystyle 1, 108; 2, 54; 3, 36; 4, 27; 6, 18; 9, 12$

The largest shared factor between 36 and 108 is 36. Therefore, 36 is the correct answer. 

Find the greatest common factor (GCF) of 56 and 64.

To find the GCF, first find the prime factors of our two numbers:

$\displaystyle 56=7*8=7*2*2*2=7(2^3)$

$\displaystyle 64=2*2*2*2*2*2=2^6$

Now, to find our GCF, we need to find the prime factors our two numbers have in common.

In this case, we have three 2's in common. In other words, we have an 8 in common. So, our GCF is 8.

Find the greatest common factor (GCF) of 324 and 56.

The GCF is the largest number which will can be evenly divided from either number.

To find the GCF, first find the prime factors of both numbers. A good trick here is to start by pulling a two out of each number until you cannot pull out a two anymore.

Beginning with 56

$\displaystyle 56\div2=28 \rightarrow 28\div2=14\rightarrow 14\div2=7$

So the prime factorization of 56 looks like:

$\displaystyle 56=2*2*2*7$

Next, let's do 324.

$\displaystyle 324\div 2=162\rightarrow162\div2=81\rightarrow 81\div 3=27$

$\displaystyle 27\rightarrow27\div3=9\rightarrow9\div3=3$

So for 324 we get:

$\displaystyle 324=2*2*3*3*3*3$

Now, the GCF can be found by taking the prime factors which are common to both numbers. Now, in this case, we have two 2's in each number, and nothing else. This means that the GCF of 56 and 324 is $\displaystyle 2*2=4$

So our answer is 4