A positive integer n is divisor rich if the sum of its proper divisors is strictly greater than n. For instance, 10 is not divisor rich because its proper divisors are 1, 2, and 5, which add to 8, and 8 6> 10. Likewise 6 is not divisor rich because its proper divisors, 1, 2 and 3, add to exactly 6, not greater than 6. (A proper divisor of a positive integer n is a positive integer smaller than n that divides into n without remainder.) a) Find the smallest divisor rich number. Show how you know it is smallest. b) Prove: any integer multiple of a divisor rich number is divisor rich. c) Is there any other class of numbers for which every integer multiple is divisor rich? (Look at your proof for Part b; maybe it accomplishes more than you have claimed so far.)

Here's a question that'll make your minds stretch!

Guest Mar 15, 2017