Question

What is the remainder when \(3^{100}\) is divided by 7?

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Find the modular cycle and reduce the exponent modulo the cycle length for efficient computation of powers.

Answer 1

The Correct Option is D

Step 1: Observe the pattern of \(3^n \mod 7\).
\[ 3^1 = 3,\quad 3^2 = 9 \equiv 2,\quad 3^3 = 27 \equiv 6,\quad 3^4 = 81 \equiv 4,\quad 3^5 = 243 \equiv 5,\quad 3^6 = 729 \equiv 1 \mod 7 \] Step 2: The cycle repeats every 6 steps. So \(3^{6k} \equiv 1 \mod 7\).
Step 3: Express \(3^{100}\) as \(3^{96} \cdot 3^4\). Since \(3^6 \equiv 1\), \(3^{96} = (3^6)^{16} \equiv 1\).
Step 4: Then \(3^{100} \equiv 3^4 \mod 7\). From Step 1, \(3^4 = 81 \equiv 4 \mod 7\).
Step 5: Therefore, the remainder is \(\boxed{4}\).