The expansion of 67 in base 7 is as follows:

First, we note that 7^{2 }= 49, and then we subtract that from 67 to get the following equation:

67 (in base 10) = 49 + 18 = 7^{2} + 14 + 4 = 1 * 7^{2} + 2 * 7 + 4 * 1 = 124 (in base 7).

Now, we also have that: (37^{15} + 27 × 43) (mod 6) = 37^{15} (mod 6) + (27*43) (mod 6)

= [ 37 (mod 6) ]^{15} + 27 * [ 43 (mod 6) ] = [ 1 (mod 6) ]^{15} + 27 * [ 1 (mod 6) ]

= 1 (mod 6) + 27 (mod 6) = 28 (mod 6) = 4 (mod 6).