Take a 3 digit number 'abc'. If we add abc, cab, and bca together then the resulting sum is divisible by:
111, 37, and 3
The numbers can be written as:
abc = 100a + 10b + c
cab = 100c + 10a + b
bca = 100b + 10c + a
Adding all we get
Sum = (100a +10b + c) + (100c + 10a + b) + (100b + 10c + a)
= (100a + 10a + a) + (100b + 10b + b) + (100c + 10c + c)
Sum = 111a + 111b + 111c
Sum = 111 (a + b + c)
=37×3(a+b+c)
sum111 = (a+b+c)
sum37= 3(a+b+c)
sum3= 37(a+b+c)
The number will be divisible by 111, 37, and 3.