Question 10.

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, 

$3^1$ mod $7=3$

$3^2$ mod $7=2$

$3^3$ mod $7=6$

$3^4$ mod $7=4$

$3^5$ mod $7=5$

$3^6$ mod $7=1$

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 $3^6$ mod $7=1$

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

Hence the answer is 4.