4.3 Perfect, deficient and abundant numbers
When we decide if a number is perfect, deficient or abundant, we look at the sum of numbers a specific numbers is divisible by. If all the numbers, except the number itself is added then we can decide if the number is deficient, abundant or perfect.
Perfect Numbers
For a number to be perfect, the sum of its divisors, except for the number itself, should be equal to that specific number.
Example
We can look at 28.
28
1 |
= |
28 |
28
2 |
= |
14 |
28
4 |
= |
7 |
28
7 |
= |
4 |
28
14 |
= |
2 |
28
28 |
= |
1 |
The number 28 has 6 divisors, namely 1, 2, 4, 7, 14 and 28. Without using 28 itself, the sum of the other divisors is added.
1 + 2 + 4 + 7 + 14 = 28
28 is then considered a perfect number.
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Deficient Numbers
A deficient numbers is larger than the sumer of its divisor, except the number itself.
Example
We can look at the number 14.
14
1 |
= |
14 |
14
2 |
= |
7 |
14
7 |
= |
2 |
14
14 |
= |
1 |
The number 14 has 4 divisors, namely 1, 2, 7 and 14. The number 14 should not be included in the sum, so we get:
1 + 2 + 7 = 10
Because 10 is less than 14, we can conclude that 14 is a deficient number.
Abundant Numbers
An abundant numbers is less than the sum of its divisors.
Example
We can try the number 18.
18
1 |
= |
18 |
18
2 |
= |
9 |
18
3 |
= |
6 |
18
6 |
= |
3 |
18
9 |
= |
2 |
18
18 |
= |
1 |
The number 18 has 6 divisors, namely 1, 2, 3, 6, 9 and 18. 18 should not included in the sum and we get:
1 + 2 + 3 + 6 + 9 = 21
Because 21 is larger than 18, we can conclude that 18 is an abundant number. |
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