+177 If \(a\), \(b\), and \(c\) are positive integers satisfying \(ab+c = bc+a = ac+b = 41\), what is the value of \(a+b+c\)?
+129852 ab + c = 41
bc + a = 41
ac + b = 41
Using the last two equations
bc + a = ac + b rearrange as
bc - ac = b - a factor theleft side
c ( b - a) = b - a divide both sides by b -a
c = (b-a) / (b -a) = 1
Using the first equation
ab + 1 = 41
ab = 40
b = 40/a
Sub this into the second equation along with the value for c
(40/a) (1) + a = 41
40/a + a = 41 multiply through by a
a^2 + 40 = 41a
a^2 - 41a + 40 = 0 factor
(a - 40) ( a -1) = 0
Setting each factor to 0 and solving for a we get that
a - 40 =0 a -1 = 0
a = 40 a =1
When a =40, then b =1 and c = 1
When a = 1, b = 40 and c =1
Regardless...... a + b + c = 42
