From the final equation a + c = d, we know that either a or c must be 2 to make both sides odd.

Consider 1 + bc + d = bd, if c = 2, then left hand side is even but right is odd, therefore a=2.

Now we have 2+c=d from the final equation. --- equation(1)

Substitute this into 1 + bc + d = bd, we have 3+c = 2b. ---equation (2)

Sub (1) and (2) into a(a + b + c + d) = c(d - b), we have a quadratic equation in c, solving will give c=-2 (rejected) or c = 11.

Now we have (a,b,c,d)=(2,7,11,13)

Btw this is really a nice question! Really fun :)