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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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Examples of smaller integers, please –  lab bhattacharjee Jun 14 at 16:27
    
Is what you are asking the following: Let a, b, and c be distinct digits. What is the largest composite number that is the sum of the 6 numbers formed by these digits? –  user148884 Jun 14 at 16:32

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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 $999$.

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No repetition, hence the mention of distinct digits :-) –  OC2PS Jun 14 at 18:30

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