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The number 132 is a three digit number with three distinct digits that is equal to the sum of all two digit numbers that are made up of its three digits:


What is the largest three digit number with this property?


In the sum on the right, each digit appears twice in the tens column and twice in the units column. So if the digits are a,b and c we have


    \begin{align*} 100a+10b+c&=22(a+b+c)\\ 78a&=12b+21c\\ 26a&=4b+7c \end{align*}

If a=4 this equation becomes 104=4b+7c, but this has no solutions as the right-hand-side is at most 99 (as b and c are both at most 9. If a=3 then the equation becomes 78=4b+7c. For values of c between 0 and 9, 78-7c is a multiple of 4 only when c=2 or c=6, and only c=6 gives a value of b that is between 0 and 9 (b=9). So the largest values is \boxed{396}.

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