Inverted nine

In the hotel,, Inverted nine" each hotel room number is divisible by 6. How many rooms we can count with three-digit number registered by digits 1,8,7,4,9?

Correct result:

n =  18

Solution:

$$\begin{gathered} n_1=114 \\ {n}_2=714 \\ {n}_3=414 \\ {n}_4=174 \\ {n}_5=774 \\ {n}_6=474 \\ {n}_7=894 \\ {n}_8=144 \\ {n}_9=744 \\ {n}_{10}=444 \\ {n}_{11}=984 \\ {n}_{12}=918 \\ {n}_{13}=978 \\ {n}_{14}=198 \\ {n}_{15}=798 \\ {n}_{16}=498 \\ {n}_{17}=948 \\ {n}_{18}=888 \\ n=18 \end{gathered}$$

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