# Thread: Hectic Numbers

1. ## Hectic Numbers

Bill is fitting his house numbers to a door. Can He screw all these digits to the door to display a number which could not be divided evenly by 3? The digits are 7 3 4 1 3.

2. Originally Posted by thecoasterfreak
Bill is fitting his house numbers to a door. Can He screw all these digits to the door to display a number which could not be divided evenly by 3? The digits are 7 3 4 1 3.
The digit sum of any number made from all these digits is 7+3+4+1+3=18,
which is divisible by 3, so any number made from exactly these digits is
divisible by 3.

RonL

3. The answer is no.

I have a different reason to CaptainBlack:
If any order of the digits is divisible by three (which 73413 is) then all orders will be, as whenever you swap two digits, in the order, either they are the same, so that the whole number does not change, or you add or subtract a multiple of 3 from the whole number, which leaves it still divisible by 3, and it is possible to create any combination of the digits by swapping two at a time.

The reason swapping two of the digits adds or subtracts a multiple of 3 every time is that the difference of one digit's change to the whole number is taken from the other which will always be a series of at least one 9 * 'the difference of the two digits' and follow with any 0s for the units bellow the digit with the lower value as a number, which will always result in a multiple of 9 (and 3).