Perfect, Abundant & Deficient Numbers
Numbers can also be classified based on how their divisors add up.
This is where perfect, abundant, and deficient numbers come in. All three are based on one simple idea:
compare a number with the sum of its proper divisors
Proper divisors = all positive factors of a number except the number itself.
Example:
Factors of 10 → 1, 2, 5, 10
Proper divisors → 1, 2, 5 (we exclude 10)
1. Perfect Numbers
A number is called perfect when the sum of its proper divisors is exactly equal to the number.
Example:
6 → Divisors: 1, 2, 3
1 + 2 + 3 = 6 → Perfect
Perfect numbers are rare and balanced.
Why Perfect Numbers Are So Rare
Perfect numbers don’t appear often because the chance that divisors add up exactly to the number is very small.
Even among all known numbers, only a few perfect numbers exist — making them special and unique.
More Perfect Examples:
28
Proper divisors: 1, 2, 4, 7, 14
Sum = 28 → Perfect
496
Proper divisors: 1, 2, 4, 8, 16, 31, 62, 124, 248
Sum = 496 → Perfect
8128
Proper divisors: 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064
Sum = 8128 → Perfect
Perfect numbers get huge very quickly,
which is why they feel “special.”
2. Abundant Numbers
A number is abundant when the sum of its proper divisors is greater than the number.
Example:
12 → Divisors: 1, 2, 3, 4, 6
1 + 2 + 3 + 4 + 6 = 16, which is > 12 → Abundant
These numbers have “extra” when compared to their own value.
Abundant Numbers Grow With Size
The larger the number, the more divisors it usually has.
More divisors → higher divisor sum → more chances to become abundant.
That’s why abundant numbers become more common as numbers get bigger.
More Abundant Examples:
12
Proper divisors: 1, 2, 3, 4, 6
Sum = 16 > 12 → Abundant
18
Proper divisors: 1, 2, 3, 6, 9
Sum = 21 > 18 → Abundant
20
Proper divisors: 1, 2, 4, 5, 10
Sum = 22 > 20 → Abundant
24
Proper divisors: 1, 2, 3, 4, 6, 8, 12
Sum = 36 > 24 → Abundant
30
Proper divisors: 1, 2, 3, 5, 6, 10, 15
Sum = 42 > 30 → Abundant
We can see the pattern:
Numbers with many factors become abundant easily.
3. Deficient Numbers
A number is deficient when the sum of its proper divisors is less than the number.
Example:
8 → Divisors: 1, 2, 4
1 + 2 + 4 = 7, which is < 8 → Deficient
Most numbers (including all prime numbers) fall into this category.
Prime Numbers and Deficiency
Every prime number is automatically deficient, because its only proper divisor is 1.
This makes their divisor sum very small, which highlights how simple prime numbers are.
More Prime Examples:
Prime numbers have only one proper divisor: 1
So their sum will always be less than the number → Deficient
5
Proper divisors: 1
Sum = 1 < 5 → Deficient
11
Proper divisors: 1
Sum = 1 < 11 → Deficient
17
Proper divisors: 1
Sum = 1 < 17 → Deficient
23
Proper divisors: 1
Sum = 1 < 23 → Deficient
This is why every prime number is automatically deficient.
More Non-Prime Examples
Not only primes – many composite numbers are also deficient.
9
Proper divisors: 1, 3
Sum = 4 < 9 → Deficient
14
Proper divisors: 1, 2, 7
Sum = 10 < 14 → Deficient
21
Proper divisors: 1, 3, 7
Sum = 11 < 21 → Deficient
27
Proper divisors: 1, 3, 9
Sum = 13 < 27 → Deficient
So deficiency is the most common type.
How They Are All Connected
Even though they sound different, these three are simply results of one comparison:
A Simple Test to Guess the Type
Without fully calculating, we can often guess:
- If a number has many factors → likely abundant.
- If it is prime or has less factors → likely deficient.
- When the divisors add up with no extra or shortage → the number becomes perfect.
Quick Recap:
- Perfect → Sum equals the number
- Abundant → Sum is more than the number
- Deficient → Sum is less than the number
Practice Questions:
State whether the number is Perfect, Abundant, or Deficient.
- 8
- 12
- 6
- 20
- 28
FAQs on Perfect, Abundant & Deficient Numbers
What makes a number perfect?
A number is perfect when the sum of its proper divisors equals the number.
Example: 6 → 1 + 2 + 3 = 6
Are all prime numbers deficient?
Yes.
Prime numbers have only one proper divisor — 1. So their divisor sum is always less than the number.
Why do abundant numbers increase as numbers get larger?
Bigger numbers tend to have more divisors, and more divisors often means a larger divisor sum, making abundant numbers more common at higher values.
Is 1 perfect, abundant, or deficient?
1 is deficient because it has no proper divisors, so the sum is 0, which is less than 1.
Can a number be both abundant and perfect?
No.
A number can only belong to one category: perfect, abundant, or deficient.