What Is the LCM of 12 and 28? | Solved With Steps

The least common multiple of 12 and 28 is 84.

You’re here because you want one clear number: the smallest number that both 12 and 28 divide into with no remainder. That number is 84. It also helps to see why—so you can trust the result, reuse the method on classwork, and catch slip-ups before they cost you points.

This page walks through three reliable ways to get the least common multiple (LCM): listing multiples, using prime factors, and the GCD×LCM relationship. You’ll also see quick checks, common errors, and a few practice problems.

What LCM means in plain math

The least common multiple of two whole numbers is the smallest positive whole number that is a multiple of both numbers.

If a number is a multiple of 12, you can write it as 12×k for some whole number k. If it’s a multiple of 28, you can write it as 28×m for some whole number m. The LCM is the smallest number that can be written both ways.

Quick signs the problem is asking for an LCM

LCM shows up when you’re asked to:

• Find a “together again” time for repeating events (one repeats every 12 minutes, the other every 28 minutes).
• Add or compare fractions with different denominators (like 5/12 and 1/28).
• Find the smallest length that can be split evenly into groups of 12 and groups of 28.

How to find the lcm step by step with multiples

This method is the most visual. You list multiples of each number until you hit a match. It’s great for smaller numbers, and it doubles as a built-in check.

Step 1: List multiples of 12

Start multiplying 12 by 1, 2, 3, and so on:

12, 24, 36, 48, 60, 72, 84, …

Step 2: List multiples of 28

Do the same for 28:

28, 56, 84, …

Step 3: Pick the first match

The first number that appears in both lists is 84, so the LCM is 84.

When the numbers get larger, long lists turn into busywork. That’s when the next method earns its keep.

What Is the LCM of 12 and 28 using prime factors

Prime factor work scales well and it explains the “why” cleanly. You break each number into primes, then rebuild the smallest product that contains both sets of prime pieces.

Prime factorize 12

Break 12 down into primes:

• 12 = 2 × 6
• 6 = 2 × 3
• So 12 = 2 × 2 × 3 = 2² × 3

Prime factorize 28

Do the same for 28:

• 28 = 2 × 14
• 14 = 2 × 7
• So 28 = 2 × 2 × 7 = 2² × 7

Build the LCM from the highest powers

Take each prime that appears in either factorization, then keep the highest exponent you see:

• Prime 2: highest power is 2²
• Prime 3: appears as 3¹
• Prime 7: appears as 7¹

Multiply them:

LCM = 2² × 3 × 7 = 4 × 3 × 7 = 84

This works because any common multiple must contain enough prime factors to cover both numbers. By taking the highest powers, you cover both while keeping the product as small as possible.

If you want a short refresher on prime factor methods for LCM, Khan Academy’s lesson on least common multiple shows the same logic with clear visuals.

Fast check method with gcd and lcm

There’s a handy identity:

(12 × 28) = GCD(12, 28) × LCM(12, 28)

If you can find the greatest common divisor (GCD), you can get the LCM with one division.

Find the GCD of 12 and 28

List factors of 12: 1, 2, 3, 4, 6, 12

List factors of 28: 1, 2, 4, 7, 14, 28

The greatest factor they share is 4, so GCD(12, 28) = 4.

Compute the LCM

Multiply first:

12 × 28 = 336

Now divide by the GCD:

LCM = 336 ÷ 4 = 84

This method is fast when the GCD is easy to spot. It’s also a strong check: if your prime-factor result gives a number that doesn’t fit this identity, something went wrong.

Why 84 makes sense as the first shared stop

Here’s a simple way to build intuition. Since 12 = 4×3 and 28 = 4×7, both share a factor of 4. So you can think of the problem as lining up 3 and 7 while keeping that shared 4 in place.

The smallest number that’s a multiple of both 3 and 7 is 21. Multiply that by the shared 4 and you get 84. Same answer, less scribbling.

Multiples table you can use as a clean proof

If your teacher wants to see the “list it out” method, a small table can make the work look neat and makes the first match stand out.

Multiple #	12 × n	28 × n
1	12	28
2	24	56
3	36	84
4	48	112
5	60	140
6	72	168
7	84	196

How to verify 84 in under ten seconds

You don’t need a long proof each time. Two quick checks do the job:

• Divisibility check: 84 ÷ 12 = 7 and 84 ÷ 28 = 3, both whole numbers.
• Smallest check: any common multiple must be a multiple of 28. The multiples of 28 below 84 are 28 and 56, and neither divides by 12.

Where students mess up and how to avoid it

Most LCM mistakes come from one of these patterns:

Mixing up LCM and GCD

LCM is about the smallest shared multiple. GCD is about the largest shared factor. A quick smell test helps: LCM is never smaller than the larger number (here, 28). GCD is never larger than the smaller number (here, 12).

Dropping a prime factor in the prime method

When you build the LCM from primes, you must include every prime that shows up in either number. With 12 and 28, that means 2, 3, and 7. Miss one, and you’ll get a number that fails the divisibility check.

Taking the wrong exponent

The LCM uses the highest exponent for each prime across the two factorizations. Both numbers have 2², so you keep 2², not 2⁴ and not 2¹.

Stopping the multiples list too early

If you list multiples, keep going until you see the first match, not the first number that “looks close.” Writing them in two neat rows or using a table prevents missed matches.

How lcm helps with fractions like 5/12 and 1/28

A common classroom use is finding a common denominator. The smooth route is to use the LCM of the denominators, since it keeps the numbers smaller than picking any random common multiple.

Turn both denominators into 84

Since the LCM is 84, convert each fraction:

• 5/12 = 5×7 / 12×7 = 35/84
• 1/28 = 1×3 / 28×3 = 3/84

Now you can add, subtract, or compare them cleanly.

LCM in repeating cycles and time problems

LCM isn’t just a worksheet topic. It’s the math behind “when will these two patterns line up again?”

Say a light blinks every 12 seconds and another blinks every 28 seconds. They blink together at the start. The next time they blink together is after 84 seconds, since 84 is the first time both 12 and 28 fit evenly into the same count.

You can check it fast: 84 seconds is 7 cycles of the 12-second light and 3 cycles of the 28-second light. No leftovers.

GCD with the Euclidean algorithm in two lines

If you ever need the GCD for larger numbers, the Euclidean algorithm keeps it tidy. For 12 and 28, it looks like this:

• 28 = 12×2 + 4
• 12 = 4×3 + 0

Once the remainder hits 0, the last non-zero remainder is the GCD. Here that’s 4. From there, the LCM drops out with (12×28) ÷ 4 = 84.

Choosing the best method based on the numbers

No single technique wins every time. The table below gives a quick pick-list so you can choose a method that fits the problem in front of you.

Method	When It’s A Good Fit	What You Do
List multiples	Small numbers; you want a visual match	Write multiples until the first shared value appears
Prime factors	Medium or large numbers; you want a reusable rule	Factor both; keep each prime to its highest power; multiply
GCD relation	You can find the GCD fast	Compute (a×b) ÷ GCD(a,b)
Build from a shared factor	The numbers share a clear factor	Pull out the shared factor, then match what’s left
Euclidean GCD first	Bigger numbers; factorization feels slow	Use the Euclidean algorithm for GCD, then the relation

Practice set with answers you can check fast

Try these. Write your answer, then check the result line.

Problem 1

Find the LCM of 8 and 14.

Answer check: 56

Problem 2

Find the LCM of 9 and 12.

Answer check: 36

Problem 3

Two bells ring every 6 minutes and every 15 minutes. When do they ring together again?

Answer check: 30 minutes

One last self-check before you move on

Before you submit an LCM answer, run this mini checklist:

• Does your number divide by both originals with no remainder?
• Is it at least as large as the bigger original number?
• Can you name one smaller candidate you ruled out, so you know it’s the smallest?

For 12 and 28, the checks land cleanly: 84 divides by both numbers, it’s larger than 28, and 28 and 56 fail the divisibility test for 12.