What are the rules for 4 and 8, and why only the last few digits?

÷4: the last two digits form a multiple of 4. ÷8: the last three digits form a multiple of 8.

Because 100 is divisible by 4 and 1000 by 8, every place above them vanishes from the remainder:

• ÷4: 11,564 → look at 64 → 64 ÷ 4 = 16 → divisible. ✓
• ÷8: 74,152 → look at 152 → 152 ÷ 8 = 19 → divisible. ✓

In general 2ⁿ divides 10ᵏ exactly when k ≥ n, so ÷2 needs 1 digit, ÷4 needs 2, ÷8 needs 3, ÷16 would need 4.