Question

One direct question from the Number System was 10 to the power 100 divided by seven, candidates had to choose the correct answer for the problem.

Answer 1

Step 1: Use modular arithmetic
We are looking for:
$10^{100} \mod 7$
Step 2: Simplify the base modulo 7
$10 \mod 7 = 3$
Thus:
$10^{100} \equiv 3^{100} \mod 7$
subsection*{Step 3: Find the cyclic pattern of powers of 3 modulo 7
Calculate successive powers of $3$ modulo $7$:
$3^1 \mod 7 = 3$
$3^2 \mod 7 = 9 \mod 7 = 2$
$3^3 \mod 7 = 27 \mod 7 = 6$
$3^4 \mod 7 = 81 \mod 7 = 4$
$3^5 \mod 7 = 243 \mod 7 = 5$
$3^6 \mod 7 = 729 \mod 7 = 1$
The powers of $3$ modulo $7$ repeat every $6$ steps. This means:
$3^{6k} \equiv 1 \mod 7 \quad \text{for any integer } k.$
Step 4: Simplify $3^{100} \mod 7$
Divide $100$ by $6$ to find the remainder:
$100 \div 6 = 16 \, \text{remainder } 4.$
Thus:
$3^{100} \equiv 3^4 \mod 7$
From Step 3:
$3^4 \mod 7 = 4$
Final Answer
The remainder when $10^{100}$ is divided by $7$ is:
$\boxed{4}$