Splitting a number into blocks of three digits from the right and then alternately subtracting and adding the blocks tests for 7, 11 and 13 at once, because $1001=7\times 11\times 13$. Using this, what is the remainder when 123456789 is divided by 13?

The blocks are 123 | 456 | 789, and the alternating total taken from the right is 789 − 456 + 123 = 456. That total differs from the original number by a multiple of 1001, so it leaves the same remainder on division by 13: $456=13\times 35+1$, giving remainder 1. Long division agrees, since $13\times 9496676=123456788$. The trap 3 comes from ADDING all three blocks — 789 + 456 + 123 = 1368, which leaves 3 — and 5 is the remainder that the same total 456 leaves on division by 11, easy to read off by mistake because the one test covers three divisors. Answering 0 assumes the striking run of digits must divide neatly.