Thread: Divisor of perfect number is not perfect

1. Divisor of perfect number is not perfect

Show that no proper divisor of a perfect number is perfect.

proof. let n be perfect, and let $d_{1},d_{2},...,d_{r}$ be divisors of n. Now, $O(n) = 2n = 1 + d_{1},d_{2},...,d_{r}$.

Now, what should I do to get O(d) not equal to 2d for all ds?

2. Try proving the contrapositive: if d is perfect and n=kd is a multiple of d, then n is not perfect. To see this, notice that if d is the sum of its proper divisors, say $d=d_1+\ldots+d_r$, then $n=kd_1+\ldots+kd_r$. So n is the sum of some of its proper divisors. But n also has other divisors that are not multiples of k (for example, 1). Therefore the sum of the proper divisors of n is greater than n.