9 Digit Number Puzzle Solution

I have given you almost a week on this puzzle, now it is time for the solution. 

The key to this puzzle was divisibility rules. This puzzle has no beautiful proof similar to that of the last puzzle but has to be tackled in a logical proof by exhaustion way. However, this solution can be made a bit quicker with modular arithmetic.

The first thing I will do to help explain things is write out the number in terms of As.

a₁, a₂ , a₃ , a₄ , a₅ , a₆ , a₇ , a₈ , a₉

I then thought are there any easy numbers which can easily be filled in. Yes there is.

The number 5 can only divide into a number which has a 5 or a zero at the end.

As we have no 0s, a₅ has to be the number 5.

The next thing we can say is that because an odd number can not divide by an even number

a₂, a₄, a₆ and a₈ have to be even numbers, otherwise the even numbers couldn’t divide into them.

Therefore, we also know that a₁,a₃,a₇and a₉ have to be odd numbers as that is all that is left.

It then gets a bit harder what to do next. I then tried to find number that could divide by 6.

The divisibility rules for 6is that it should by even and all the digits in the number sum should to a multiple of3. As the first three number have to be a multiple of 3, they have to sum to three. Therefore, a₄, a₅ and a₆ have to sum to three to divide by six. We also know from above that a₄ and a₆ have to be even too.

You then need to run through all the possible combinations that fit those rules. It turns out that there are three number that can be a₄, a₅, a₆. Those are 258 or 654.

Next it is easiest to find to find a₇ and a₈. As a₆ is either an 8 or 4, if a₇,a₈ divides by 8 itself so will  the whole number up to a₈. Also a₁, a₂ , a₃ have to divide by three. With these rules and the odd and even placing we established before we have a set of possible 9 digit numbers which fit most of the rules. These are:

183654729

189654327

189654723

381654729

741258963

789654321

981654327

981654723

987654321

The last thing we need to do is to try dividing every number by 7, the only number we have not factored in yet, because it has a very complicated divisibility rule.

Only one number is divisible by 7 and that is 381654729 which is the solution. That is it!