I've done that (just summing all the listed proper divisors and checking ).Euclid’s theorem aboutperfectnumbers*depends on the prime divisor property, which will be proved in the next exercise.

Assuming this for the moment, it follows that if is aprime,

then the proper divisors of are and .

Given that the divisors ofare those just listed, show thatisperfectwhenisprime.

That if is

*prime, then isperfect.

Help with this one.We can now fill the gap in the proof of Euclid’s theorem on perfect numbers*(above exercise), using the prime divisor property.

Use the prime divisor property* to show that the proper divisors of , for any odd prime , are and .

*If is a prime that divides , then divides or .