N is a 4-digit positive integer such that the sum of the four digits of N equals the product of the first two digits of N and also equals the product of the last two digits of N.

Find all N's and prove there are no others.

Solution:

1. It
is easily found that there cannot exist
an abcd solution with three or four digits distinct.

2.
A generic **abab** solution will generate the

** abba, baba** , ** baab** solutions. <br>

3. Write 2*(a+b)-a*b
as **a =2*b/(b-2),** which produces integer values for b=3 or 4 or
6 **only**.

4.
We get generic answers: (a,b ) = (3,6); (4,4 ); (6,3 ).

5.The
set of all valid answers is: (**4,****4,4,4); (3,6,6,3);** ** ****(3,6,3,6); ****(6,3,6,3); );(6,3,3,6);**

**General remark: **** Finding **__all__ Ns constitutes

**a full proof there **** ****are no others. **