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.
Correct Answer: 4
Solution and Explanation
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:
\[\boxed{4}\]
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