A divisibility test for 7 that is actually usable
Tests for 2, 3, 5, 9 and 11 are well known. The one for 7 is usually skipped because the textbook version is awkward. Here is a version that is easy to do in your head.
Split off the last digit, double it, and subtract it from the rest. Repeat until the number is small.
3,976 → 397 − 12 = 385
385 → 38 − 10 = 28 = 4 × 7 ✓ divisible
1,234 → 123 − 8 = 115
115 → 11 − 10 = 1 ✗ not divisible
Why it works
Write the number as 10a + b. Then 10a + b ≡ 0 (mod 7) exactly when
a − 2b ≡ 0 (mod 7), because multiplying a − 2b by 10 gives
10a − 20b = (10a + b) − 21b, and 21 is a multiple of 7. Since 10 and 7 share no factors,
nothing is lost in the step.
The same trick works for 13 if you add four times the last digit instead of subtracting twice.