When $$10^{100}$$is divided by 7, the remainder is

To find the value of $$10^{100}mod\left(7\right)$$
When 10 is divided by 7, it leaves a remainder 3, so the above equation can be written as, 
$$3^{100}mod\left(7\right)$$

Now looking at the cyclicality of powers of 3 when divided by 7, 
$$3^1mod  7=3$$

$$3^2mod  7=2$$

$$3^3mod  7=6$$

$$3^4mod  7=4$$

$$3^5mod  7=5$$

$$3^6mod  7=1$$

From this calculation, it is evident that the powers of 3 modulo 7 repeat every 6 steps. This forms a cycle: 3, 2, 6, 4, 5, 1

$$3^{100}=\left(3^6\right)^{16}\times\ \left(3^4\right)$$

Since $$3^6mod 7=1$$

We just need to consider $$3^4mod 7$$ which equals 4

Hence the answer is 4.