# Least common multiples

The concept of the **least common multiple (LCM) **can be expected as part of the GMAT and GRE quant sections. There is an easy way to calculate a LCM, which we will explain here.

Let’s imagine you are asked what the least common multiple of **4**, **9** and **27** is.

First, break each number down into its prime factors.

**4** can be broken down into **2 x 2**, or **2 ^{2}**.

**9** can be broken down into **3 x 3**, or **3 ^{2}**.

**27** can be broken down into **3 x 3 x 3**, or **3 ^{3}**.

Next, take the highest count of each prime factor and multiply these together. We have **2 ^{2}**,

**3**, and

^{2}**3**to consider. The highest count of the prime factor

^{3}**2**is

**2**. The highest count of the prime factor

^{2}**3**is

**3**(which is higher than

^{3}**3**).

^{2}Therefore, to get the LCM, multiply **2 ^{2} x 3^{3} = 4 x 27 = 108**.

Now that you have this concept down, consider the following question from GMAT Tutor's question bank:

*What is the least common multiple of two distinct integers, each greater than 67?*

68

69

136

68

^{2}68 x 69

As soon as you see the term “least common multiple,” you may be tempted to adopt the approach above. It will form a good basis for solving, but this question also requires a bit more.

We know the two integers are different and they must both be greater than **67**. If we are looking for the LCM, it is tempting to think of the two smallest integers which are greater than **67**, namely **68 **and **69**. If we break these down into their prime factors, we have:

**68**: **2 x 34 = 2 x 2 x 17 = 2 ^{2} x 17**

**69: 13 x 13 = 13 ^{2}**

The LCM of **68 **and **69 **would therefore be **22 x 17 x 13 ^{2}**, which would be quite a large number.

However, consider two numbers which are not the smallest integers greater than **67**. A hint may come from the answer choices. Look at *option C*. What about the two integers **68 **and **136**? The LCM of these two numbers is actually **136**, because **136** is just **2 x 68**. C it is.

Having the right method to calculate a LCM is important. When trickier questions pop up, you can call upon that basic method to help find the solution more quickly.