*begin background*

For any positive integer n, define d(n) to be n plus the sum of the digits of n.

For example, d(79) = 79 + 7 + 9 = 95.

Take integer n as a starting point, you can construct the infinite increasing sequence of integers n, d(n), d(d(n)), d(d(d(n))), ....etc

For example, if you start with 33, the next number is 33 + 3 + 3 = 39, the next

is 39 + 3 + 9 = 51, the next is 51 + 5 + 1 = 57, and so you generate the sequence : 33, 39, 51, 57, 69, 84, 96, 111, 114, 120, 123, 129, 141, ...

The number n is called a generator of d(n). In the sequence above, 33 is a generator of 39, 39 is a generator of 51, 51 is a generator of 57, and so on. Some numbers have more than one generator: for example, 101 has two generators, 91 and 100.

A number with no generators is a self-number.

There are thirteen self-numbers less than 100: 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, and 97.

*end background*

Now come my questions:

a.(d2) What is the smallest n making a **10 ^{n}** a self number?

b.(d4) Checking integers with 1 in the beginning, 1 in the end and n-1 zeros between the ones (i.e.

**10**what value of n creates a number with 3 generators ?

^{n}+1)c.(d5 or d4 after a hint) What is the smallest number with 4 generators ?