Perfect Number Checker
Check whether a number is perfect, abundant or deficient by comparing it with the sum of its proper divisors.
Last reviewed by the Radiatus Cloud team
Check whether a number is perfect, abundant or deficient.
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Classify a number by its divisors
The proper divisors of a number are all its positive divisors except itself. This checker adds them up and classifies the number accordingly: it is perfect if the sum equals the number, abundant if the sum is greater, and deficient if the sum is smaller. Twenty-eight is a perfect number because its proper divisors, one, two, four, seven and fourteen, add up to exactly twenty-eight. The famous smallest perfect number is six.
The tool lists the divisors and their sum so you can see how the classification is reached.
Perfect numbers through history
Perfect numbers have fascinated mathematicians since antiquity, and remarkably all known perfect numbers are even and connected to special primes called Mersenne primes. Whether any odd perfect number exists is a famous open question. Abundant numbers, whose divisors overshoot, and deficient numbers, which fall short, make up the rest of the integers, with deficient numbers being by far the most common.
Perfect numbers are extremely rare; after six and twenty-eight, the next are four hundred and ninety-six and eight thousand one hundred and twenty-eight. All calculation happens locally in your browser.
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Frequently Asked Questions
What is a perfect number?
A number equal to the sum of its proper divisors, such as six, whose divisors one, two and three add up to six.
What are abundant and deficient numbers?
Abundant numbers have proper divisors that sum to more than the number, while deficient numbers sum to less. Most numbers are deficient.
Are there odd perfect numbers?
None are known, and whether any exist is a famous unsolved problem. Every perfect number found so far is even.
What are the first few perfect numbers?
Six, twenty-eight, four hundred and ninety-six and eight thousand one hundred and twenty-eight. They are very rare and grow quickly.
Privacy & Security
Everything runs in your browser; nothing is uploaded.
How to Use
Enter a positive integer to classify it by its divisors.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.