If a, b, and c are positive integers satisfying ab + c = bc + a = 13 and ac + b = 41, what is the value of a + b + c?
+128475 ab + c = 13
bc + a = 13
ac + b = 41
The first two equations imply that
ab + c = bc + a
ab - bc = a - c
b ( a - c) = a - c divide both sides by a - c
b = 1
Looking at the last equation
ac + 1 = 41
ac = 40
Let a = 8 and c = 5
These values make the first two equations true
So
a + b + c = 8 + 1 + 5 = 14
