Greatest common factor (GCF) & Least common multiple (LCM)

Greatest common factor (GCF)

The greatest common factor (GCF) of a pair of numbers is the largest number that divides both numbers evenly. To find the GCF of two numbers, list the factors of each number, identify the factors they share, and choose the greatest one. Note that the GCF uses all factors, not just prime factors.

Here’s a table showing the factors of 148 and 212.

The numbers 148 and 212 share the bolded factors: 1, 2, and 4. The greatest shared factor is 4, so the GCF is 4.

There is no shortcut to finding the greatest common factor of any two numbers - you’ll always have to list all the factors (a.k.a. divisors) of the two numbers and determine the largest shared factor. Let’s try a question.

What is the greatest common factor of $x^2{y}^{3}z$ and $x^3{y}^2{z}^5$, given that $x$, $y$, and $z$ are all different prime numbers?

A. $xy{z}^4$
B. $x^6{y}^6{z}^5$
C. $x^5{y}^5{z}^6$
D. $x^2{y}^{2}z$
E. $xyz$

This question uses the same idea as the integer example. Instead of listing every factor, you can build the GCF by taking only the parts that must appear in both expressions.

Try solving this question on your own, and then keep reading for our explanation.

The GCF of $x^2{y}^{3}z$ and $x^3{y}^2{z}^5$ is the largest expression that divides both.

Start by pulling out the largest power of $x$ that appears in both expressions. Both contain at least $x^2$:

• $x^2{y}^{3}z=x^2(y^{3}z)$
• $x^3{y}^2{z}^5=x^2(x{y}^2{z}^5)$

So $x^2$ is part of the GCF. Now compare what remains: $y^{3}z$ and $x{y}^2{z}^5$.

Next, pull out the largest power of $y$ that appears in both. Both contain at least $y^2$:

• $y^{3}z=y^2(yz)$
• $x{y}^2{z}^5=y^2(x{z}^5)$

So $y^2$ is also part of the GCF. Now compare what remains: $yz$ and $x{z}^5$.

Finally, check for a shared factor of $z$. Both contain at least one $z$:

• $yz=z(y)$
• $x{z}^5=z(x{z}^4)$

So $z$ is part of the GCF. At this point, the remaining factors are $y$ and $x{z}^4$. The problem tells you that $x$, $y$, and $z$ are different prime numbers, so there are no additional shared factors.

Multiply the shared factors together:

$$\text{GCF}(x^2{y}^{3}z,x^3{y}^2{z}^5)=x^2{y}^{2}z$$

Least common multiple (LCM)

The least common multiple (LCM) of two numbers is essentially the opposite of the greatest common factor (GCF). Whereas the GCF is the greatest shared factor of the two numbers, the LCM is the smallest number that is a multiple of both. Finding the least common multiple is especially useful when adding or subtracting fractions with different denominators.

To find the least common multiple of a set of numbers, prime factor each number, take the greatest number of each prime factor that appears in any one factorization, and multiply those primes together. Here’s an example of finding the LCM of $9$ and $78$.

First, prime factor these numbers:

• $9=3\ast 3$
• $78=2\ast 3\ast 13$

Next, for each prime, keep the largest count you see in either factorization:

• $2$ - One $2$ needed (zero 2s in 9, one 2 in 78)
• $3$ - Two $3$s needed (two 3s in 9, one 3 in 78)
• $13$ - One $13$ needed (zero 13s in 9, one 13 in 78)

Now multiply them together:

$$\text{LCM}(9,78)=2\ast 3\ast 3\ast 13=234$$

Let’s see how this comes into play in a real question.

Solve for $x$.

$2/5+13/10+12/45=x/30$

You can use the LCM to help you solve this question. Give it some thought and try to answer it.

There are almost always multiple ways to solve a math equation. The “naive” approach is to multiply by the product of all denominators so every fraction becomes an integer. That would require multiplying by $5\ast 10\ast 45\ast 30=67500$, which is much larger than necessary.

Instead, use the LCM of the denominators $5$, $10$, $30$, and $45$.

We can find it by prime factoring:

• $5=5$
• $10=2\ast 5$
• $30=2\ast 3\ast 5$
• $45=3\ast 3\ast 5$

Collecting the greatest quantity of each prime factor per number gives us the following:

• 1x $2$ (from 10 or 30)
• 2x $3$ (from 45)
• 1x $5$ (they all have one 5)

Multiplying these together gives:

$$\text{LCM}(5,10,30,45)=2\ast 3\ast 3\ast 5=90$$

If you rewrite each fraction with denominator $90$, you can add them directly. You do this by multiplying each fraction by a form of $1$ that changes the denominator to $90$.

For instance, $90/5=18$, so multiply $2/5$ by $18/18$ to get $36/90$. This doesn’t change the value of the fraction because $18/18=1$.

$$\begin{array}{rl}2/5+13/10+12/45& =x/30\\ 2/5\ast (18/18)+13/10\ast (9/9)+12/45\ast (2/2)& =x/30\ast (3/3)\\ 36/90+117/90+24/90& =3x/90\\ 177/90& =3x/90\\ 177& =3x\\ 59& =x\end{array}$$

Although you can solve this specific question without using the LCM method, it’s important to understand how the method works and to practice it. The GRE test writers craft questions that require these techniques; e.g. they might use numbers that are too large to be multiplied together naively with the calculator.