# Long Division Logic Problem

I know some of you enjoy some good math logic problems (Myself included) here is one I had fun with that took me a bit of time to solve. The A in these numbers is a number 0-9, it is the same number of all the A's in the puzzle. The question marks can be any other number 0-9 except whatever you chose as A (example. if you picked A to be 0 then none of the question marks are 0). The question marks do not need to be the same number, could for example be 1827 for the top of the puzzle, with 2 being the A.

Hope you are practiced up on your long division! Pulling apart the structure above,


?A? x {2-9} = ??AA
?A? x {2-9} = ??A
?A? x A = ?A??
?A? x {2-9} = ????

Where {2-9} is a single digit (can't be 0 or 1).
A is also not 1 or 0 or whatever number is used for {2-9}.

The first term is the most interesting, if we just look at the
tens and ones terms we end up with: A? x {2-9} = AA

I found four cases where this is true:
37 x 9 = 333 (A = 3 and, ?A? = ?37, etc)
43 x 8 = 344 (A = 4 and, ?A? = ?43, etc)
48 x 3 = 144 (A = 4 and, ?A? = ?48, etc)
84 x 7 = 588 (A = 8 and, ?A? = ?84, etc)

So A is 3, 4, or 8.

There must be a number {2-9} such that ?AY = ??A (Y known),
and ALSO multiplying ?A? x A results in a four digit number
so that number {2-9} must be less than A.

?X? can not be ?37 (9 x 7 is 63 but 9 is not less than 3).
?X? can not be ?43 (either 2 nor 3 times ?43 can be ??4).

?X? can be ?48 (3 times ?48 can be ??4). ?X? can be ?84 (2 and 7 times ?84 can be ??8).
So A is 4, or 8.

In order for ?48 to work, there needs to be a number N such that (N48 x 4 = ?4??) and N48 x 3 = ??4 ... and there is no such number.
So A is 8.

N84 x 8 = ?8?? N=4 are N=9 are the only numbers which make this work.
N84 x (2 or 7) = ??8 (2 or 7 because of the ending 8)
In order to have 3 digits N must be 4.

?A? x A = ?A?? is actually: 484 x 8 = 3872
?A? x ? = ??A is actually: 484 x 2 = 968
?A? x ? = ???? is actually: 484 x (3,4,5,6,9) (because it can't have any 8s).

So, ?A? x ??A? = ????A?? (the entire problem, is...)
484 x 728N = ????8?? (with N = 3,4,5,6, or 9)
N = 9 because that's the only number with an 8 there.
So the entire problem is: 484 x 7289 = 3527876

• Thank you Mithrandir... and interesting. Did you click on one button to do all that or was it line by line? – Dark Matter Mar 26 '17 at 21:33