Perfect, Abundant, and Deficient Numbers Explained (With Practice Problems)

Every whole number is built from smaller factors, but something interesting happens once we add those factors together. The total can fall short of the number, land right on it, or climb past it.

Mathematicians use those three outcomes to sort every whole number into one of three groups. We call them perfect, abundant, and deficient.

Today, we’ll introduce all three groups and show how to classify any number.

What Are Proper Divisors of a Number?

Proper divisors of a number are any positive whole numbers that divide into it evenly, except for the number itself.

We can think of it as finding all the factors that build a number, then leaving the full-sized number off the final list.

So, let's look at 20 and see what factors it has: 

• The factors of 20 are 1, 2, 4, 5, 10, and 20.

• Now we drop the 20 itself. 

• What's left, 1, 2, 4, 5, and 10, are the proper divisors.

We can find the proper divisors of a number in three simple steps.

1. List all the factors of the number.

2. Remove the actual number from the list.

3. Keep the rest of the factors.

Our tutors start every lesson on perfect, abundant, and deficient numbers here, with proper divisors. 

Now that we can find proper divisors, we can see how these three groups compare.

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What Is a Perfect Number?

A perfect number is a positive whole number that equals the exact sum of its proper divisors. 

Let's see this in action with the number 6.

First, we need to find its proper divisors.

• Find the factors. The divisors of 6 are 1, 2, 3, and 6.

• Remove the number itself.

• We drop 6, leaving the proper divisors 1, 2, and 3.

So, to prove that our number here is perfect, we need to check if the proper divisors add up to it.

Let's check.

1 + 2 + 3 = 6.

Because the sum of its proper divisors equals the original number, 6 is a perfect number!

Here is a fun fact.

6 is the smallest perfect number, and the first on the list of the 52 perfect numbers we know of. And guess what? Mathematicians have been searching for them for more than 2,000 years.

If you'd like to learn more about the history behind these numbers, check out our full article on perfect numbers.

What Is an Abundant Number?

An abundant number is a positive whole number whose proper divisors add up to more than the number itself.

Let's use the number 12 as an example and look for its proper divisors.

• Find the factors. The divisors of 12 are 1, 2, 3, 4, 6, and 12.

• Remove the number itself.

• We drop 12, leaving the proper divisors 1, 2, 3, 4, and 6.

Now we add. 

1 + 2 + 3 + 4 + 6 = 16.

As we can see, 16 is greater than 12, and that extra amount is why we call 12 abundant. 

Abundance is the difference between the sum of a number's proper divisors and the number itself. For 12, the abundance is 4, since 16 − 12 = 4.

Abundant numbers also connect directly to perfect numbers. If we take any perfect number and multiply it by a whole number greater than 1, the result is always abundant. 

Let's test it out. 

We know that 6 is a perfect number. So if we multiply 6 by 2, we get 12. 

The name suggests these numbers are very common, but there are only 22 of them in the first 100 numbers. The next group we'll cover is actually the largest of the three by far.

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What Is a Deficient Number?

A deficient number is a positive whole number, and its proper divisors add up to less than the number itself.

Let’s see what happens with 8. 

• First, we find the factors. The divisors of 8 are 1, 2, 4, and 8.

• Remove the number itself.

• We drop 8, leaving the proper divisors 1, 2, and 4.

Now we add. 

1 + 2 + 4 = 7.

7 is less than 8, and that's exactly what makes 8 a deficient number. 

Deficiency measures how much less the sum of a number's proper divisors is than the number itself. For 8, the deficiency is 1, since 8 − 7 = 1.

What is interesting is that prime numbers take this pattern to its extreme.

We already know that every prime number has exactly two factors, 1 and itself. So when we remove the number itself, we're always left with just 1. So, what can we assume from this?

We can assume that every prime number is also a deficient number.

Let’s take 13 as a prime number example. 

The divisors of 13 are 1 and 13. We drop 13 itself, leaving 1 as the only proper divisor. We know that 1 is less than 13, which makes it a deficient number.

As we mentioned, deficient numbers also happen to be the most common of the three by a wide margin. Among the first 100 numbers, 76 are deficient, compared to 22 abundant numbers and just 2 perfect numbers.

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What's the Difference Between Deficient, Abundant, and Perfect Numbers?

The difference between these three groups comes down to one comparison. 

We add up a number's proper divisors, then check how that sum lines up against the number itself. That single check is all it takes to sort any whole number into one of the three groups.

The table below lays out all three side by side, along with the symbol we use for each comparison. 

Notice that each row uses numbers we already worked through earlier in this guide. The table lays them out side by side so we can easily compare how each category behaves.

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Practice: Classify These Six Numbers

Ready for some practice?

For each number below, find its proper divisors, add them up, and decide whether the number is perfect, abundant, or deficient. Work through each one the same way we did earlier in this guide.

1. 18

2. 28

3. 10

4. 20

5. 14

6. 17

Check your answers at the bottom of the guide.

Mathnasium uses personalized learning plans and hands-on teaching techniques to help students think more deeply about numbers and develop number sense.

How Mathnasium Helps Students Master Any Math Concept

Mathnasium is a math-only learning center empowering students of all skill levels to learn and master math.

Our tutors guide students through everything from foundational factor rules to advanced concepts like number classifications. We make sure every student leaves our center with a complete understanding of each topic.

To help build that level of understanding, we use the Mathnasium Method™, our proprietary teaching approach, designed around individual students’ needs and learning styles. 

Here is how it works.

• Diagnostic Assessment and Personalized Learning Plans: Every student begins with a diagnostic assessment that identifies both visible skill gaps and the reasoning patterns behind them. From that starting point, we build a personalized learning plan tailored to their needs and goals, helping them build lasting math mastery at a pace that works for them.

• Teaching for Understanding: Our specially trained tutors use plain, everyday language and a mix of verbal, visual, mental, tactile, and written techniques so concepts land in a way that makes sense to each student.

• Problem-Solving and Critical Thinking: When a concept feels challenging, we break it down into manageable parts and guide students through both the how and the why. Over time, this builds the problem-solving skills and critical thinking they can use in math and everyday life.

• An Engaging and Fun Learning Environment: Sessions are designed to keep students motivated and enjoying the process. We celebrate every bit of progress, and that consistent recognition builds confidence with each session. Over time, students develop a more positive relationship with math and greater confidence in their own abilities.

The results reflect that approach.

• 94% of parents report improvement in their child's math skills and understanding.

• 93% of parents report an improved attitude toward math after attending Mathnasium.

• 90% of students saw improvement in their school grades.

We operate over 1,100 centers across North America, bringing our proven approach to communities everywhere.

Families in and around Mansfield, TX, can visit Mathnasium of Mansfield North, a trusted local center with a proven record of building confident math thinkers.

Whether your student is looking to catch up, keep up, or get ahead in math, our local team is happy to help!

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Pssst! Check Your Answers Here

Here's how each number breaks down.

1. 18 → 1 + 2 + 3 + 6 + 9 = 21 → abundant

2. 28 → 1 + 2 + 4 + 7 + 14 = 28 → perfect

3. 10 → 1 + 2 + 5 = 8 → deficient

4. 20 → 1 + 2 + 4 + 5 + 10 = 22 → abundant

5. 14 → 1 + 2 + 7 = 10 → deficient

6. 17 → only proper divisor is 1 → deficient