The Least Common Multiple of 24 and 28 — And Why You Actually Need to Know It
Let’s cut right to it: the least common multiple of 24 and 28 is 168.
But if you’re here because you’re staring at a homework problem or trying to figure out why this matters at all, you’re not alone. On top of that, most people learn how to find the LCM once, forget it immediately, and then panic when they see it again. Here's the thing — it’s not just busywork. It shows up everywhere, from simplifying fractions to syncing schedules.
Not the most exciting part, but easily the most useful.
So let’s break it down. Not just the answer, but the why and the how. In real terms, because real talk? Memorizing the steps without understanding them is how you end up stuck in math class forever.
What Is the Least Common Multiple?
The least common multiple (LCM) of two numbers is the smallest number that both of them divide into evenly — no remainders, no decimals, just clean division Small thing, real impact..
Think of it like this: you’re planning two events. One happens every 24 days, the other every 28 days. When will they line up again? Worth adding: that’s the LCM. In this case, day 168.
Prime Factorization — Your Best Friend
The cleanest way to find the LCM is through prime factorization. Here’s how it works:
For 24:
- 24 = 2 × 12
- 12 = 2 × 6
- 6 = 2 × 3
- So, 24 = 2³ × 3¹
For 28:
- 28 = 2 × 14
- 14 = 2 × 7
- So, 28 = 2² × 7¹
Now here’s the trick — for each prime number that shows up, take the highest power of that prime from either number:
- For 2: the highest power is 2³ (from 24)
- For 3: the highest power is 3¹ (from 24)
- For 7: the highest power is 7¹ (from 28)
Multiply them together: 2³ × 3¹ × 7¹ = 8 × 3 × 7 = 168
That’s your LCM.
Why Does This Matter?
Honestly, this is the part most guides get wrong. They treat LCM like a puzzle with no real-world application. But it’s not just about passing a test.
Fractions — The #1 Use Case
When you add or subtract fractions with different denominators, you need a common denominator. Even so, the least common denominator? That’s the LCM of the denominators.
Say you’re adding 1/24 and 1/28. Day to day, without finding the LCM, you’d probably multiply the denominators (24 × 28 = 672) and call it a day. Which means that works — but it gives you unnecessarily large numbers. The LCM (168) keeps things clean.
Convert:
- 1/24 = 7/168
- 1/28 = 6/168
- 7/168 + 6/168 = 13/168
See how much nicer that is than dealing with 672?
Real-World Scheduling
Ever notice how traffic lights, maintenance schedules, or shift rotations eventually sync up? That’s the LCM in action And that's really what it comes down to..
If one machine needs servicing every 24 hours and another every 28 hours, they’ll both need service at the same time every 168 hours — that’s exactly one week. Knowing this helps you plan ahead instead of getting caught off guard.
How to Find the LCM — Multiple Methods
There’s more than one way to skin this cat. Here are the three most common approaches:
Method 1: Prime Factorization (What We Just Did)
This is the gold standard. It works for any pair of numbers, no matter how big. The steps are simple:
- Break each number into its prime factors
- For each prime, take the highest power that appears
- Multiply those together
It’s systematic, reliable, and scales well Simple, but easy to overlook..
Method 2: Listing Multiples
This one’s old-school but intuitive. Just list out multiples of each number until you find a match:
Multiples of 24: 24, 48, 72, 96, 120, 144, 168, 192... Multiples of 28: 28, 56, 84, 112, 140, 168.. Which is the point..
Boom — first match is 168.
But here’s the catch: this gets unwieldy fast. Try this with 47 and 63. You’ll be listing multiples until next week.
Method 3: Using the GCF (Greatest Common Factor)
There’s a neat relationship between LCM and GCF:
LCM(a, b) = (a × b) / GCF(a, b)
First, find the GCF of 24 and 28. The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24. The factors of 28 are: 1, 2, 4, 7, 14, 28. The greatest common one is 4.
Now plug it in:
- LCM(24, 28) = (24 × 28) / 4 = 672 / 4 = 168
This method is fast if you already know the GCF. But finding the GCF can be just as much work as finding the LCM directly.
Common Mistakes — What Most People Get Wrong
I know it sounds simple — but it's easy to miss these traps That's the part that actually makes a difference..
Mixing Up LCM and GCF
People constantly swap these two. Here's the thing — the LCM is the smallest number that both divide into evenly. The GCF is the largest number that divides both evenly. They’re related but opposite in a way Simple, but easy to overlook. Practical, not theoretical..
For 24 and 28:
- GCF = 4 (the biggest number that divides both)
- LCM = 168 (the smallest number both divide into)
Forgetting to Use the Highest Power
This is the #1 error with prime factorization. When you break down 24 and 28:
- 24 = 2³ × 3
- 28 = 2² × 7
Some people multiply all the primes together: 2³ × 2² × 3 × 7 = 8 × 4 × 3 × 7 = 672. That’s wrong.
You only use the highest power of each prime. So for 2, you use 2³ (not 2²). The correct calculation is 2³ × 3 × 7 = 168.
Assuming the Product Is Always the LCM
A lot of people just multiply the two numbers: 24 × 28 = 672. e.Plus, the LCM is always less than or equal to the product, and only equals the product when the two numbers share no common factors (i. On top of that, that is a common multiple — but it’s not the least one. , they’re coprime) That's the whole idea..
Not obvious, but once you see it — you'll see it everywhere.
Since 24 and 28 share common factors (2 and 4), their LCM is smaller than their product.
Practical Tips — What Actually Works
Here’s what I wish someone had told me when I was learning this:
Tip 1: Master Prime Factorization First
If you’re shaky on breaking numbers into primes, LCM will feel like guesswork. Practice with smaller numbers first. Once it clicks, the rest follows naturally Simple, but easy to overlook..
Tip 2: Use the GCF Shortcut When It’s Obvious
If you can quickly spot the GCF, the formula LCM(a, b) = (a × b) / GCF(a, b) is lightning-fast. For numbers like 24 and 28, where the GCF is small and obvious
, this method saves time. But don’t force it — if finding the GCF takes longer than listing multiples, switch methods.
Tip 3: Know When to Stop Listing Multiples
You don’t need to go forever. A good rule of thumb: if you’ve listed more than 10 multiples without a match, try another method. Your time is better spent switching strategies than grinding through endless lists.
Tip 4: Double-Check with the Relationship
Once you think you’ve found the LCM, verify it:
LCM(a, b) × GCF(a, b) = a × b
For 24 and 28:
168 × 4 = 672
24 × 28 = 672 ✅
This quick check catches most errors That alone is useful..
Wrapping It Up
Finding the LCM isn’t about memorizing steps — it’s about understanding relationships between numbers. Whether you prefer listing multiples, using prime factorization, or leveraging the GCF formula, the key is picking the right tool for the job and avoiding common pitfalls.
Remember:
- LCM = smallest shared multiple
- GCF = largest shared factor
- Prime factorization is your foundation
- Check your work using the LCM × GCF = a × b relationship
With practice, you’ll instinctively reach for the most efficient method instead of defaulting to brute-force listing. And that’s when LCM problems go from frustrating to straightforward.
Whether you're simplifying fractions, solving word problems, or just building number sense, mastering the LCM pays off. The next time you see 24 and 28, you’ll know exactly what to do — and you’ll get 168 without breaking a sweat.
