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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?

 May 11, 2021
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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

 

 

cool cool cool

 May 11, 2021

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