Question

Fill in the blank:
Two perfect numbers are \[\_\] and \[\_\].

Answer 1

Hint: To find the perfect numbers, we should know what really a perfect number is. A perfect number is a positive integer that is equal to the sum of all the positive divisors of the number, except the number itself. So, add the divisors or factors of positive integer excluding that number. If the sum is equal to that number, then the number is perfect.

Complete step-by-step solution:
We know that a perfect number is a positive integer that is equal to the sum of all the positive divisors of the number, except the number itself. That is if \[n \in {Z^ + }\], where \[{Z^ + }\] is a set of positive integers, and \[1,a,b,c,n\] are the factors of \[n\]. Then if \[1 + a + b + c = n\], then \[n\] is a perfect number.
For ex., factors of \[6\] are\[1,2,3,6\]. If we add those factors except \[6\] we get \[1 + 2 + 3 = 6\] , that is the number itself. So, \[6\] is a perfect number.
Same way , factors of \[28\] are \[1,2,4,7,14,28\]. If we add those factors except \[28\], we get \[1 + 2 + 4 + 7 + 14 = 28\], that is the number itself. So, \[28\] is a perfect number.
So, we fill in the blanks as Two perfect numbers are \[\underline 6 \] and \[\underline {28} \].

Note: To find the perfect number, beside hit and trial method there is also a formula given by Euclid. Euclid approximately over two thousand years ago, had shown that all even perfect numbers can be represented by, $N = 2p-1(2p -1)$ where p is a prime for which 2p -1 is a Mersenne prime. Some of the perfect numbers are \[6,28,496,8128\,etc.\]