Remainder of a Sum Divided by 7

$\Large{70005 = 7\times 10^4 + 5 \equiv 5 \pmod{7} \\ 7004 = 7\times 10^3 + 4 \equiv 4 \pmod{7} \\ 703 = 7\times 10^2 + 3 \equiv 3 \pmod{7} \\ 72 = 7\times 10 + 2 \equiv 2 \pmod{7} \\ 1\equiv 1 \pmod{7}} \\ \large{\therefore 70005+7004+703+72+1 \equiv 5+4+3+2+1 \equiv 15 \equiv \huge{\boxed{1}} \pmod{7}}$

A simple solution:

1. Add the ones digits:

5+4+3+2+1 = 15

1. Divide the sum by 7, the remainder is the same as that of the sum of the original addends:

15/7 = 2.14285714285.....

0.142857....*7 = 0.999999.... = 1

70005=70000+5 7004=7000+4 703=700+3 72=70+2 Thus, 70005+7004+703+72+1= 77770 + 14 +1