Question

check whether 6n can never end with the digit 2 for any natural number n

Answer 1

6 x   2 =   12
6 x   7 =   42
6 x 12 =   72
6 x 17 = 102

For any 'n' equal to ( 2 + any multiple of 5), 6n ends with the digit '2' .

The conjecture appears to be pretty well shot.