Find all the prime numbers (a, b, c) that abc = 5 (a + b + c)
We know that Right hand side is a multiple of 5. This means that on the LHS atleast one of the 3 numbers a, b or c should have a factor of 5. Now all of the 3 numbers a, b and c are prime. This means that one of them should be 5 itself.
Let a = 5
Now the equation becomes
5bc = 5 . (5 + b + c)
=> b.c = 5 + b + c (cancelling 5 from both the sides)
=> c.(b - 1) = 5 + b
=> c.(b - 1) = 5 + (b - 1) + 1
=> c.(b - 1) - (b - 1) = 5 + 1
=> (c - 1).(b - 1) = 6
The LHS is a product of 2 different integers and RHS is 6. This means that the 2 numbers on LHS are either (3 x 2) or (6 x 1).
Case-1 : (c - 1).(b - 1) = 3.2 = 6
This gives c = 4 and b = 3 (This is not possible because c like a and b should be a prime. 4 is not a prime number).
Case-2 : (c - 1).(b - 1) = 6.1 = 6
This gives c = 7 and b = 2 (Here both b an c are prime numbers as required. Hence this is a right solution).
Therefore the 3 prime numbers (a, b, c) are (5, 2, 3) Notice that the order of a,b and c does not matter