Joining Numbers

If we have 2 numbers \(\overline { abc }\) and \(\overline { def }\) we can join this 2 number to make other more large \(\overline { abcdef }\). If we join more one number \(\overline { ghi}\) we have \(\overline { abcdefghi }\).

We can reduce a number summing their digits to get only a 1-digit number: \(362 \Rightarrow 3+6+2=11 \Rightarrow 1+1=2\). If we reduce the number 362 we get 2.

If we join all 3-digit numbers multiples of 9 to make a other more large, and reduce this number, we get a number \(n\). What is the value of \(\frac {n}{3}\)?

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