Number Theory
# Linear Diophantine Equations

Solve the following cryptarithm:

\[\begin{array} { l l l l l l } & & S & E & N & D \\ +& & M & O & R & E \\ \hline & M & O & N & E & Y \\ \end{array}\]

and find the value of \(S+E+N+D+M+O+R+Y.\)

I am thinking of a four digit positive integer with distinct digits.

Of course, there's a total of \(4!-1=23\) ways to rearrange the digits to form a new 4 digit positive integer.

If the sum of these other 23 numbers is 157193, what is the number that I was thinking of?

Find the number of ordered pairs of positive integer solutions \((m, n)\) for \[20m + 12n = 2012.\]