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Can You Find a Perfect Number?

Stage: 2 and 3
Article by NRICH team

People have been searching for number patterns since ancient times. Mathematicians noticed that some numbers are equal to the sum of all of their factors (but not including the number itself).

6 is a number that equals the sum of its factors: 1 + 2 + 3 equal 6.
Numbers like 6 that equal the sum of their factors are called perfect numbers.
6 is the first perfect number .

4 is not a perfect number because the sum of its factors (besides 4 itself), 1+2, is less than 4. Numbers like 4 are known as deficient numbers . What does the word deficient mean? Why do you think numbers having factors adding to less than the number itself are called deficient numbers?

12 is not a perfect number because the sum of its factors, 1+2+3+4+6 is greater than 12. Numbers like 12 are known as abundant numbers . What does the word abundant mean? Why do you think numbers having factors adding to more than the number itself are called abundant numbers?

Since the time of Pythagoras (about 500 BC), mathematicians have tried to find as many perfect numbers as they can. By 1999 only 38 perfect numbers had been found.

Can you find the next perfect number after 6?


Could a perfect number also be a prime number?


Do you think there are more perfect, deficient or abundant numbers?


Add the factors of all the numbers up to 50. Was your guess correct?


Can you find any patterns in your results?