Form a number from the digits 0 to 9 such that the first digit is divisible by one, the first two digits form a number that is divisible by two, and so forth. Prove also that this number is unique.
Let each of the digits in the number be represented by a letter so that the number is: abcdefghij
First of all, we can see that ab/2, abcd/4, abcdef/6, abcdefgh/8, and abcdefghij/10 are integers, therefore, b, d, f, h, and j are all even. Since we only have five even digits (0, 2, 4, 6, and 8), a, c, e, g, and i are all odd digits. Furthermore, knowing that certain parts of the number are divisible by certain numbers gives us the following constraints:
Constraint 9 tells us nothing since constraint 10 states that j is 0, and the sum of the remaining digits is divisible by nine.
From constraints 4 and 5, the only possible values for the digits d and e are
de = 25 or 65.
Combining constraints 3 and 6 we see that (d+e+f)/3 must be an integer and that f is even. Thus,
de = 25 implies f = 8.
de = 65 implies f = 4.
This gives us two possible cases:
def = 258 or 654.
Using constraint 8, we can see that
def = 258 implies gh = 16 or 96.
def = 654 implies gh = 32 or 72.
Now we have four possibilities that satisfy constraints 4, 5, 6, and 8:
defgh = 25816, 25896, 65432, or 65472.
Notice that in all 4 cases, 2 and 6 are used. Thus b = 4 or 8.
Now using constraint 3 with the above constraints leaves us with a number of possibilities for abc:
abc = 147, 183, 189, 381, 741, 789, 981, or 987.
Even with the above constraints, this gives us ten possibilities for the first eight digits:
abcdefgh = 14725896, 18365472, 18965432, 18965472, 38165472, 74125696, 78965432, 98165432, 98165472, or 98765432.
However, we can use constraint 7, and see that only
abcdefg = 3816547 is divisible by 7.
Thus the final answer is 3816547290. Furthermore, this is a unique answer.
This problem can also be stated for numbers with less than ten digits. We will not go into the details of a complete solution for these situations, but will list the answers below. We will leave the solution (using the constraints stated above) up to the reader.
