Divisibility by 9
Proof:
Let the number be abcd.
Remainderabcd can be written as a×1000 + b×100 + c×10 + d which can also be written as a×(9×111+1) + b×(9×11+1) + c×(9+1) + d which can also be written as a×(9×111)+ a + b×(9×11)+ b + c×(9)+c + d Every power of 10 i.e. 10^{x} when divided by 9 has remainder 1. So a×(9×111)+ b×(9×11)+ c×(9) is divisible by 9 as it has 9 as factor or common factor. The number is divisible by 9 if a+b+c+d is divisible by 9.
Examples
