By six

From the digits 1,2,3,4 we create the long integer number 123412341234. .. .. , which will have 962 digits. Is this number divisible by 6?

Result

x =  1

Solution:

n=962 n1=n/4=962/4=4812=240.5 n2=⌊n1⌋=⌊240.5⌋=240 n3=n−4⋅ n2=962−4⋅ 240=2  s=n2⋅ (1+2+3+4)+1+2=240⋅ (1+2+3+4)+1+2=2403 s3=s/3=2403/3=801  12341234...123412 x=1n = 962 \ \\ n_{ 1 } = n/4 = 962/4 = \dfrac{ 481 }{ 2 } = 240.5 \ \\ n_{ 2 } = \lfloor n_{ 1 } \rfloor = \lfloor 240.5 \rfloor = 240 \ \\ n_{ 3 } = n-4 \cdot \ n_{ 2 } = 962-4 \cdot \ 240 = 2 \ \\ \ \\ s = n_{ 2 } \cdot \ (1+2+3+4) + 1 + 2 = 240 \cdot \ (1+2+3+4) + 1 + 2 = 2403 \ \\ s_{ 3 } = s/3 = 2403/3 = 801 \ \\ \ \\ 1234 1234 ... 1234 12 \ \\ x = 1

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