Perfect Numbers
A number equal to the sum of its proper divisors.
Key idea
A number equal to the sum of its proper divisors.
In depth
A perfect number equals the sum of its proper divisors (those smaller than itself). They are rare: 6, 28, 496, 8128. Even perfect numbers come from Mersenne primes: 2^(p−1)(2^p − 1).
Worked example
6: 1 + 2 + 3 = 6. 28: 1 + 2 + 4 + 7 + 14 = 28.
Formula
Real-world uses
- A historical curiosity studied in number theory and recreational maths puzzles
Common mistake
Including the number itself in the divisor sum.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Perfect Numbers?
A number equal to the sum of its proper divisors. It is a topic at the advanced level in Arithmetic & Number Sense.
What is the formula for Perfect Numbers?
The key formula is 6 = 1+2+3.
Where is Perfect Numbers used in real life?
Real-world examples: A historical curiosity studied in number theory and recreational maths puzzles.
What should I learn before Perfect Numbers?
Make sure you are comfortable with Roman Numerals, the previous topic in Arithmetic & Number Sense.
What is a common mistake with Perfect Numbers?
Including the number itself in the divisor sum.
How can I practise Perfect Numbers?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Perfect Numbers?
Move on to Triangular Numbers next.
Related advanced topics in Arithmetic & Number Sense
- Repeating Decimals – Decimals with a repeating block are rational and convertible to fractions.
- Compound Interest – Interest earned on principal and past interest.
- Inverse Proportion – When one quantity doubles, the other halves.
- Rational Numbers – Numbers expressible as p/q with integers p, q, q≠0.
- Irrational Numbers – Real numbers that cannot be written as a fraction, such as √2 and π.
- Real Numbers – All rational and irrational numbers; the continuous number line.