Åk 6–9

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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.

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.