# Prime Numbers

Topic:
Numbers

## Prime Numbers

﻿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