What is the largest number (so, yes, I am looking for a discrete integer not an algebraic expression) by which the sum of all 3-digit numbers formed with the non-zero, distinct digits a, b, and c MUST be divisible?
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If digits cannot be repeated, the sum of all the numbers is $2(a+b+c)(111)$. For each of the digits occurs twice in each position.
Since only $1$ divides all possible $a+b+c$, where $a$, $b$, $c$ range over the non-zero digits, the largest number that divides all the sums is $222$.
If repetition of digits is allowed, the sum is $9(a+b+c)(111)$, and the largest number is $999$.