How To Take Lcm Of Three Numbers

Stop Reaching for the Calculator — Here's How LCM of Three Numbers Actually Works

You've probably stared at a problem asking for the least common multiple of three numbers and immediately started listing multiples by hand. In real terms, 24, 48, 72, 96... and your patience slowly drains away. There's a better way.

The least common multiple (LCM) of three numbers isn't some mysterious beast that requires a calculator app or a lucky guess. It's a logical process once you know what you're doing. And yeah, it shows up in real math problems, fraction work, and even scheduling scenarios you might not expect.

What Is LCM, Really?

The LCM of a set of numbers is the smallest number that all of them divide into evenly. No remainders. No fractions. Just clean division.

For two numbers, this is straightforward enough. But when you throw in a third, things can feel messier. Here's the thing — the good news? Still, the core idea stays exactly the same. You're still looking for that one number — the smallest one — that all three original numbers divide into without leaving a remainder.

Think of it like this: if you had three runners on different lap schedules, the LCM would tell you when they all cross the starting line at the same time again.

Why Three Numbers Changes Everything

With two numbers, you might get away with listing multiples until you find a match. But with three? On the flip side, that list gets long fast. And the chance of missing the least* common multiple — instead of just a common multiple — gets real.

This is where prime factorization saves the day.

Why It Matters Beyond the Classroom

LCM isn't just busywork your teacher made up to ruin your day. It has practical uses.

When you're adding or subtracting fractions with different denominators, you need a common denominator. The LCM of those denominators gives you the least* common denominator — which keeps your numbers smaller and your arithmetic simpler.

It also shows up in real-world timing problems. If one event happens every 6 days, another every 8 days, and a third every 12 days, the LCM tells you when all three line up.

Understanding how to find the LCM of three numbers means you're not stuck guessing or relying on a calculator that might not even have the right function.

How to Find the LCM of Three Numbers

There are two reliable methods. Pick whichever clicks for you.

Method 1: Prime Factorization (The Reliable Way)

It's the method that scales. It works for three numbers, four numbers, or however many you throw at it.

Step 1: Break each number into its prime factors.

Let's use 12, 18, and 20 as our example.

• 12 = 2 × 2 × 3
• 18 = 2 × 3 × 3
• 20 = 2 × 2 × 5

Step 2: Identify every prime number that appears.

Looking at our factorizations, the primes involved are 2, 3, and 5.

Step 3: For each prime, take the highest power that appears in any single factorization.

• For 2: the highest power is 2² (from 12 and 20)
• For 3: the highest power is 3² (from 18)
• For 5: the highest power is 5¹ (from 20)

Step 4: Multiply those together.

LCM = 2² × 3² × 5¹ = 4 × 9 × 5 = 180

So the LCM of 12, 18, and 20 is 180.

Method 2: Division Method (The Step-by-Step Way)

This one feels more like following a recipe, but it's just as valid.

Step 1: Write the three numbers in a row.

12, 18, 20

Step 2: Divide by the smallest prime that divides at least one of them.

Start with 2. It divides all three.

• 12 ÷ 2 = 6
• 18 ÷ 2 = 9
• 20 ÷ 2 = 10

Write the quotients below.

Step 3: Repeat with the new row.

Again, divide by 2 (it still divides two of the three numbers).

• 6 ÷ 2 = 3
• 9 ÷ 2 = 9 (doesn't divide evenly, so carry it down)
• 10 ÷ 2 = 5

New row: 3, 9, 5

Step 4: Continue until no common prime divides any two numbers.

Now divide by 3.

• 3 ÷ 3 = 1
• 9 ÷ 3 = 3
• 5 ÷ 3 = 5 (carry down)

New row: 1, 3, 5

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Divide by 3 again.

• 1 ÷ 3 = 1 (carry down)
• 3 ÷ 3 = 1
• 5 ÷ 3 = 5 (carry down)

New row: 1, 1, 5

Now divide by 5.

• 1 ÷ 5 = 1 (carry down)
• 1 ÷ 5 = 1 (carry down)
• 5 ÷ 5 = 1

Final row: 1, 1, 1

Step 5: Multiply all the divisors together.

LCM = 2 × 2 × 3 × 3 × 5 = 180

Same answer. Different path.

Common Mistakes People Make

Here's where most people trip up — and honestly, it's usually the same few errors.

Mixing Up LCM with GCD

The greatest common divisor (GCD) and the least common multiple (LCM) are related but opposite. GCD is about what divides into* your numbers. Now, lCM is about what your numbers divide into*. Confusing them leads to answers that are way too small or way too big.

Forgetting to Use the Highest Power

In the prime factorization method, it's tempting to just multiply all the primes you see. But that's not right. Day to day, you need the highest* power of each prime that appears in any of the factorizations. Using a lower power gives you a number that one of your originals won't divide into evenly.

Stopping Too Early

With the division method, people often stop dividing once one number becomes 1. But you keep going until all numbers are 1. Stopping early means you've missed factors, and your LCM will be too small.

Listing Multiples Instead

This works for two numbers if you're patient, but with three? Day to day, you're basically gambling. Which means the LCM could be 50, or 500, or 5,000. Listing multiples by hand for three numbers is inefficient and error-prone.

Practical Tips That Actually Work

Use Prime Factorization for Larger Numbers

If your numbers are big — say, 48, 72, and 108 — listing multiples is going to take forever. Here's the thing — prime factorization cuts through the noise. Break them down, grab the highest powers, multiply. Done.

Double-Check Your Answer

Whatever method you use, plug your LCM back in. Think about it: divide it by each of your original numbers. If you don't get clean division every time, something went wrong.

Remember the Relationship

For any two numbers, LCM(a, b) × GCD(a, b) = a × b. This doesn't directly extend to three numbers, but knowing the relationship helps you catch obvious errors.

Practice with Friendly Numbers First

Start with small sets like 4, 6, and 8. That's why build your confidence. Then work up to trickier combinations. Muscle memory matters here.

FAQ

Can the LCM of three numbers be one of the original numbers?

Yes. Day to day, if one number is a multiple of the other two, it's automatically the LCM. To give you an idea, the LCM of 6, 12, and 24 is 24.

What if one of the numbers is 1?

Then the LCM is just the LCM of the other two numbers, since 1 divides into everything.

**Is there a shortcut for

three numbers?**

Not a universal one, but if you notice a pattern—like if two of the numbers are prime or if one number is a direct multiple of another—you can skip the heavy lifting. Always check for these "shortcuts" before diving into a full prime factorization.

Can the LCM be smaller than the numbers provided?

No. Day to day, by definition, the Least Common Multiple must be equal to or greater than the largest number in your set. If you get a result smaller than your starting numbers, you've accidentally calculated the GCD.

Conclusion

Mastering the Least Common Multiple is less about memorizing complex formulas and more about understanding the "DNA" of numbers through their prime factors. Whether you prefer the systematic approach of the division method or the precision of prime factorization, the goal is the same: finding the smallest shared destination for your numbers.

By avoiding the common pitfalls—like stopping the division process too early or confusing the LCM with the GCD—you turn a potentially tedious math problem into a quick, logical exercise. Remember to always double-check your work by dividing your result by your original numbers; if the math holds up, you’ve found your answer. Keep practicing, and soon, finding the LCM will become second nature.