Positive Integers

$ab + c = bc + a\\ ab - bc = a-c \\ b(a-c) = a-c \\ b(a-c) - (a-c) = 0\\ (a-c)(b-1)=0\\ a=c\\ b=1$

Therefore, a and c must equal to 6.5, though the question states that all of these numbers are integers.

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?

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b = 1

ac + 1 = 41

ac = 40

a + c = 13

(40/c) + c = 13

c = 8,   c = 5

a + b + c = 14

1 * (5 + 8) = 13

1 + (5 * 8) = 41