100 digit number % 7

Follow this equation : $\frac{1}{7}(\overline{\ldots a_7 a_6 a_5 a_4 a_3 a_2 a_1 a_0})$ $= \frac{1}{7}(10^0 a_0 + 10^1 a_1 + 10^2 a_2 + 10^3 a_3 + 10^4 a_4 + 10^5 a_5 + 10^6 a_6 + 10^7 a_7 + \ldots)$ $= \frac{1}{7}a_0 + (1+\frac{3}{7})a_1 + (14+\frac{2}{7})a_2 + (142 + \frac{6}{7})a_3 + (1428 + \frac{4}{7})a_4 + (14285 + \frac{5}{7})a_5 + (142857 + \frac{1}{7})a_6 + (1428571 + \frac{3}{7})a_7 + \ldots$ $= \frac{1}{7}(a_0 +a_6 + \ldots) + \frac{3}{7}(a_1 + a_7 + \ldots) + \frac{2}{7}(a_2 + a_8 + \ldots) + \frac{6}{7}(a_3 + a_9 + \ldots) + \frac{4}{7}(a_4 +a_10 + \ldots) + \frac{5}{7}(a_5 + a_11 + \ldots)$ That's how the digit define the remainder. From that equation, we know that if [a 0 + a 6 + \ldots = a 3 + a 9 + \ldots\ = n] It will make those (0+6k) and (3+6k) number cannot control remainder. I don't know how to explain, but here i find the remainder : 739,289 mod 7 --> 5 0 0 , 0 5 0 --> 5(5/7) + 5(3/7) --> 5(8/7) --> 40/7 -- >5 + 5/7 ---another example--- 295,381,629,618,492,628 mod 7 --> 12 20 16 , 15 11 17 --> 0 9 0 , 3 0 1 --> 9(4/7) + 3(2/7) + 1(1/7) --> 36/7 + 6/7 + 1/7 --> 43/7 --> 6 + 1/7

That is manual way. You also can solve with calculator. If it asked 391,294,163,813,713,017,371,824,817 mod 7 Then in calculator, calculate 391,294,163 mod 7, then it will result 1. Then add the result in front of next number, calculate 1,813,713,017 mod 7, then it will result 4. Then add the result in front of next number, calculate 4,371,824,817 mod 7, then it will result 3, the remainder of 27 digit above divide 7.

That's what i can say. About the problem, there is some tricky number which make the problem easier as 4 digit ;)