jmc

An integer is divisible by $18$ if and only if the sum of its digits is divisible by $9$ and the last digit is even (meaning it is divisible by both 9 and 2). The sum of the digits of 142857 is 27, so this number is a multiple of 9. The number is odd, so it is not a multiple of 18. Instead the number is 9 more than a multiple of 18. Therefore $$\boxed{9}\equiv 142857(\mathrm{mod}18).$$