Find a six digit integer Find an integer with six different digits such that the six digit integer is divisible by each of its digits.
For example, find ABCDEF such that A, B, C, D, E and F all can divide the number ABCDEF.
Show your answer along with mathematical reasoning.
 A: The number $861432$ will work.  I guess $123864$ looks nicer.
A: $4\\
24\\
624\\
3624\\
183624$
2,4,8 are good to work with, because once you find three digits at the end, you can put whatever you want in the front.
3,6,9 are good to work with too, because if you find something that divides by 3 or 9, you can shuffle the digits and it still divides by 3 (or 9).
and 1 is just a no-brainer
A: Here's a software solution written in the c language using (obscure?) bit-wise logic checking if a number fulfills the criterion or not.
int check_number(int a){
  int l = a, mask = 0;
  for(;l;l/=10) {
    mask |= (1<<(l%10));
    mask |= (!(mask%2)) && (a%(l%10) != 0);}
  return (__builtin_popcount(mask)==6) && (!(mask%2));
}

The loop runs over each decimal digit by doing a sequence of dividing (/) by 10 and calculating the remainder (%). As you can see for book-keeping we revisit the bit field representation and population count operation in this question, but in this solution each logical value means that one digit is present AND divides our number. We could save a line if we were to find a cleverer way to avoid division by 0.
The result conforms with @Eric Tower s solution for at least smallest and largest numbers (123648 and 984312). I haven't bothered to check all of them.
A: The highest match is 984312
I wrote a quick Python3 script to find all the matches:
for number in range(100000, 1000000):
    string = str(number)
    digits = []
    unique = True

    for ch in string:
        if ch in digits:
            unique = False
            break
        else:
            digits.append(ch)

    if not unique: # If the digits are not unique, move to the next iteration
        continue

    if all([False if digit == '0' else number % int(digit) == 0 for digit in str(number)]):
        # Number divides perfectly with every digit
        print(number)

This code will print every working number, each to a new line.
A: The LCM of $1,2,3,4,5$ is $60$.
$$\frac{134520}{60} = 2242$$
