A three-digit number is evenly divisible by 7 if and only if twice its first digit added to the number formed by its two last digits gives a result that’s divisible by 7. So, for example, 938 is divisible by 7 because 2 × 9 + 38 = 56 = 7 × 8.

In fact this can be extended to numbers of any length: 229187 → 2 × 2291 + 87 = 4669 → 2 × 46 + 69 = 161 → 2 × 1 + 61 = 63 = 7 × 9.

(J. Kashangaki, “A Test for Divisibility by Seven,” Mathematical Gazette 80:487 [March 1996], 226.)