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 $20m + 12n = 2012.$