Creating a number based on given conditions

Question

In the grade 7 math curriculum here in Alberta we teach divisibility. I'm trying to write a program to create questions based on the idea of a student being given say, 4 digits, and then choosing one of their own to create say, "the largest number divisible by 5".

I have a few simple ideas. Get a list of 4 numbers...

RandomInteger[{0, 9}, 4]

Arrange them

Permutations[list]

Turn them into numbers...

FromDigits /@ Permutations[list]

I'm sure there must be an elegant way to use Mathematica to do this. I'd need to have the original 4 digits and the answer so I could create the question under program control.

I.E. Given the digits 9, 0, 0, 2 and 1 other digit of your choice, create the largest 5 digit number you can that is divisible by 5.

I'd appreciate any hints or suggestions.

Answer 1

Something like this?

f[initialNbrOfdigits_, extradigits_, divisor_] := Module[{sol = -Infinity, s},
   While[sol == -Infinity, 
         s   = RandomInteger[{0, 9}, initialNbrOfdigits];
         sol = Max@Select[FromDigits /@  Flatten[Permutations[Join[s, #]] & /@ 
                              Tuples[Range[0, 9], {extradigits}], 1], Divisible[#, divisor] &]];
   Column@{Row[{"Given the digits ", s, " and ", extradigits, 
                " more of your choice, create the largest ", extradigits + initialNbrOfdigits, 
                " digit number you can that is divisible by ", divisor}],
           Row[{"Solution: ", sol}]}
   ];
f[3, 2, 773]

Given the digits {9,9,4} and 2 more of your choice, create the largest 5 digit number you can that is divisible by 773

Solution: 98944

Edit

With a small modification to avoid considering numbers with leading zeroes, you could ask also for the minimum:

f[initialNbrOfdigits_, extradigits_, divisor_, f_] := 
  Module[{sol = -Infinity, s}, 
   While[sol == -Infinity, 
    s = RandomInteger[{0, 9}, initialNbrOfdigits];
    sol = f@Select[FromDigits /@ Flatten[Permutations[Join[s, #]] & /@ 
                   Tuples[Range[0, 9], {extradigits}], 1], Divisible[#, divisor] && # >= 
                                                   10^(extradigits + initialNbrOfdigits - 1) &]];
   Column@{Row[{"Given the digits ", s, " and ", extradigits, 
                " more of your choice, create the " , f, " ", extradigits + initialNbrOfdigits, 
                " digit number you can that is divisible by ", divisor}], 
           Row[{"Solution: ", sol}]}];

f[3, 2, 18, Min]

Given the digits {8,2,2} and 2 more of your choice, create the Min 5 digit number you can that is divisible by 18
Solution: 12258