Question 431642
1. Suppose the number is {{{10a + b}}}, where {{{a <> b}}}. Then, {{{a + b + ab = 10a + b}}} --> {{{a + ab = 10a}}} --> {{{1 + b = 10}}}, b = 9. We can assume {{{a}}} to be as small as possible, so 19 is the smallest such number.


2. 28, because the sum of its proper divisors is 1+2+4+7+14 = 28. In fact, all even perfect numbers are in the form {{{(2^(n-1))(2^n - 1)}}} where {{{2^n - 1}}} is prime. This is because the sum of divisors function is a multiplicative function for relatively prime integers. When {{{n = 3}}}, {{{2^3 - 1 = 7}}}, prime, so {{{(2^2)(2^3 - 1) = 28}}}, a perfect number.