$6341723110832864 \bmod 2 = 0$

$6341723110832864 \bmod 3 = (6341723110832860 + 4) \bmod 3 = ((6341723110832860 \bmod 3) + (4 \bmod 3)) \bmod 3 = (1+1)\bmod 3 = 2$

so

$6341723110832860 \bmod 6 = (3*0 - 2*2) \bmod 6 = -4 \bmod 6 = 2$

similarly

$6341723110832864 \bmod 4 = (6341723110832800 + 64) \bmod 4 =$

$((6341723110832800 \bmod 4) + (64 \bmod 4)) \bmod 4 = 0+0 \bmod 4 = 0$

so

$6341723110832864 \bmod 12 = (4*2 - 3*0) \bmod 12 = 8$