What 2 Numbers Multiply To Get 72

Ever sat staring at a math problem that felt unnecessarily frustrating? It’s one of those moments where the numbers seem to dance around the page, refusing to settle into a neat little equation. And you know the feeling. You know the answer is there—it’s hiding somewhere in the simple logic of multiplication—but your brain just isn't clicking with it.

If you're currently stuck on the question of what two numbers multiply to get 72, you aren't alone. It’s a common hurdle, whether you're a student trying to finish homework before dinner or an adult trying to mentally calculate a tip or a discount.

What Is the Logic Behind Finding Factors?

When we ask what numbers multiply to get 72, we are really looking for the factors of 72. In plain English, we want to find pairs of whole numbers that, when combined through multiplication, result in that specific total.

Think of it like breaking a large block into smaller, equal-sized pieces. If you have 72 bricks, how many ways can you stack them into perfect rectangles? You could have one long row of 72, or you could have several rows with an equal number of bricks in each.

The Concept of Factor Pairs

The most helpful way to look at this is through factor pairs. A factor pair is just a set of two numbers that work together. Here's one way to look at it: 8 and 9 are a pair because $8 \times 9 = 72$. But they aren't the only ones.

If you find one pair, you've actually found two. If you know that $2 \times 36 = 72$, you automatically know that $36 \times 2 = 72$. This symmetry is a great shortcut when you're working through a list.

Prime vs. Composite Numbers

To really understand how 72 is built, you have to look at its "DNA." In math, we call this prime factorization. Some numbers are "prime," meaning they can't be broken down any further (like 2, 3, 5, or 7). Other numbers are "composite," meaning they are built by multiplying those prime numbers together.

72 is a highly composite number. But it has a lot of building blocks. If you break 72 down to its most basic elements, you get $2 \times 2 \times 2 \times 3 \times 3$. Every single pair of numbers that multiplies to 72 is just a different way of grouping those specific prime numbers together.

Why Does Finding These Numbers Matter?

You might be thinking, "Why am I spending time on this? I have a calculator for a reason." It’s true, but there's more to it than just getting the answer.

Simplifying Fractions

If you've ever dealt with fractions in algebra or even in cooking measurements, you've needed this skill. If you have a fraction like 72/84 and you want to simplify it, you need to find the Greatest Common Factor (GCF). Knowing the ways 72 can be broken down allows you to see those connections instantly.

Scaling and Proportions

In real-world scenarios—like construction, woodworking, or even graphic design—you often deal with ratios. Worth adding: if you need to scale a design up or down, you're essentially playing with factors. If a pattern repeats every 6 units and you have 72 units of space, knowing that 6 is a factor of 72 tells you immediately that the pattern will fit perfectly without any leftover space.

Mental Math and Speed

There is a certain "brain fitness" aspect to this too. And being able to quickly identify that 72 is divisible by 6 or 8 makes you faster at mental arithmetic. It builds a sense of number sense, which is that intuitive feeling for how numbers relate to one another. Once you stop seeing 72 as just a random digit and start seeing it as a collection of possibilities, math becomes much less intimidating.

How to Find All the Pairs for 72

So, how do you actually do it without losing your mind? Now, you don't want to just guess randomly. You need a system. The best way to ensure you haven't missed anything is to work through the numbers in order, starting from 1.

The Systematic Division Method

Here is the most reliable way to find every single pair:

1. Start with 1: Every number is divisible by 1. So, $1 \times 72 = 72$.
2. Move to 2: Since 72 is an even number, 2 must work. $2 \times 36 = 72$.
3. Try 3: A quick trick for 3 is to add the digits ($7 + 2 = 9$). Since 9 is divisible by 3, 72 is too. $3 \times 24 = 72$.
4. Check 4: Does 4 go into 72? Yes, $4 \times 18 = 72$.
5. Check 5: Numbers divisible by 5 must end in 0 or 5.72 ends in 2, so skip it.
6. Check 6: Since 2 and 3 both worked, 6 will work. $6 \times 12 = 72$.
7. Check 7: 7 doesn't go into 72 evenly ($7 \times 10 = 70$, leaving a remainder of 2).
8. Check 8: We know our multiplication tables! $8 \times 9 = 72$.
9. Stop at the Square Root: Once you reach a number where the pair is already listed (in this case, 9 is the partner to 8), you can stop. You've found them all.

The Complete List of Factor Pairs

To make it easy, here is the full list of pairs that multiply to 72:

For more on this topic, read our article on what is the oxidation number of nitrogen in no2 or check out is a single bond a sigma bond.

• 1 and 72
• 2 and 36
• 3 and 24
• 4 and 18
• 6 and 12
• 8 and 9

Dealing with Negative Numbers

Here is something most people forget: multiplication rules apply to negatives too. If the question doesn't specify "positive integers," there are actually an infinite number of pairs if you include negative numbers.

Here's one way to look at it: $-8 \times -9 = 72$. If you multiply one positive and one negative, you get a negative result, so those won't help you reach a positive 72. Or $-2 \times -36 = 72$. But in most standard math problems, we stick to the positive whole numbers.

Common Mistakes / What Most People Get Wrong

I've seen people struggle with this for years, and it usually comes down to a few specific errors.

Missing the Middle Pairs

The most common mistake is jumping straight from 4 to 8 and forgetting about 6. Now, people often assume that if a number doesn't "look" obvious, it isn't a factor. But 6 is a very common factor for 72, and it's easy to skip it if you're rushing.

Confusing Factors with Multiples

Basically a big one. Here's the thing — a factor is a number that goes into* 72. In real terms, a multiple is a number that 72 goes into*. In practice, * Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. * Multiples of 72: 72, 144, 216, 288... and so on.

If you're working on a problem and you're stuck, double-check that you aren't accidentally looking for multiples when you should be looking for factors.

Forgetting the "1" and the Number Itself

It sounds silly, but when people are in a rush to find "the" numbers, they often overlook the most obvious pair: 1 and the number itself. It’

Forgetting the "1" and the Number Itself

It sounds silly, but when people are in a rush to find "the" numbers, they often overlook the most obvious pair: 1 and the number itself. On top of that, it's easy to dismiss these as trivial, but they are just as valid as any other factor pair. Remember, 1 divides evenly into every integer, and every number is a multiple of itself.

Not Stopping at the Right Time

Some students continue checking numbers well past the square root, wasting time and effort. Others stop too early, missing valid pairs. The key is knowing that once you've checked up to the square root of your target number, you've found every possible factor pair. For 72, since √72 ≈ 8.49, checking up to 8 is sufficient.

Why This Matters: Real-World Applications

Understanding how to find factor pairs isn't just busywork for a math class. This skill has practical applications:

• Simplifying Fractions: When reducing fractions like 72/108, knowing that both numbers share common factors (like 36) makes the process much faster.
• Organizing Groups: If you need to arrange 72 people into equal groups, knowing the factor pairs tells you all your options: 8 rows of 9, 6 rows of 12, etc.
• Mental Math: Recognizing that 72 = 8 × 9 helps you quickly calculate problems like 72 ÷ 12 by thinking "8 × 9 ÷ 12 = 8 × (9÷12) = 8 × 0.75 = 6."

Conclusion

Finding factor pairs is a foundational skill that combines pattern recognition, division facts, and logical thinking. Now, by following a systematic approach—starting small, checking divisibility rules, and stopping at the right point—you can efficiently identify all factor pairs without missing any. Worth adding: remember to consider negative factors when appropriate, avoid common pitfalls like confusing factors with multiples, and don't forget the obvious pairs. With practice, what initially seems like a tedious process becomes a quick and reliable tool for solving more complex mathematical problems.