Let us begin with these pieces:

- The number is less than the # of years in a century...

- There are 2 digits in the whole part..

- There are 5 digits all together...

- The number is greater than 5 dozen...

From this given, we can establish that:

Now, let us translate the rest of the given into algebraic terms:

1. Now, step number 1: solve simultaneously:

to get

2. Now solve simultaneously again with:

to get

3. Then, let us examine the given:

It is obvious to see that d = 0. It is impossible that c = 0 since the product of c and e is 24. This is the key; plug in the relations we got previously:

4.

If c = 6, then e = 4 but this doesn't satisfy the inequality

Therefore, c = 4 and e = 6