Divisibility in base $7$ problem Find all integers between $0 \leq a \leq 2400$ such they are divisible by $8$ and that their base 7 development has at least $3$ equal digits.

In base 7 a number is divisible by 8 if and only if the alternating sum of its digits is divisible by 8. 
If 3 digits are equal, and the alternating sum is divisible by 8, the 4th digit must also be equal.
Therefore, the answers are 0000, 1111, 2222, 3333, 4444, 5555, and 6666 (all written in base 7). (I am not sure if you would characterize 0000 as having 3 equal digits or not).
