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.
Finding all N's
