Factor Sums

Not counting itself, the number 6 has the factors 1, 2, and 3, which add to 6. The number 28 has the same property (its factors are 1, 2, 4, 7, and 14). Can you come up with a three-digit number that has this property? What about a four-digit number? Click below for the answers.

If you knew that a number that is the sum of its own proper divisors is called a Perfect number, this puzzle was pretty easy. You could just search for that name and find the solutions are 496 and 8,128. Perfect numbers have been known at least as far back as Euclid (323–283 BCE), who included a formulation for then in his book of Elements.

The formulations states that $q(q + 1) / 2$ is a perfect number whenever $q$ is a prime of the form $2^p - 1$ for prime $p$ (now known as a Mersenne prime). So, if we know the first few Mersenne primes, we can calculate the first few perfect numbers.

$3(3 + 1) / 2 = 6$
$7(7 + 1) / 2 = 28$
$31(31 + 1) / 2 = 496$
$127(127 + 1) / 2 = 8,128$
$8,191(8,191 + 1) / 2 = 33,550,336$

The ancient Greek mathematicians would not have known that 8,191 was a prime, so Euclid would only have known the first four Perfect numbers. Now you can say you know something that Euclid didn't!