Joining Numbers

If we have 2 numbers a b c \overline { abc } and d e f \overline { def } we can join this 2 number to make other more large a b c d e f \overline { abcdef } . If we join more one number g h i \overline { ghi} we have a b c d e f g h i \overline { abcdefghi } .

We can reduce a number summing their digits to get only a 1-digit number: 362 3 + 6 + 2 = 11 1 + 1 = 2 362 \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 n . What is the value of n 3 \frac {n}{3} ?

3 1 4 2

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