What is the remainder when 3 raised to the power of 102 is divided by 10?

What is the remainder when 3 raised to the power of 102 is divided by 10?

• A. 7
• B. 9
• C. 3
• D. 1

Answer: (B)

To find the remainder when \(3^{102}\) is divided by \(10\), we can utilize patterns in the powers of \(3\) modulo \(10\). Calculating the first few powers of \(3\): - \(3^1 = 3\), remainder when divided by \(10\) is \(3\). - \(3^2 = 9\), remainder when divided by \(10\) is \(9\). - \(3^3 = 27\), remainder when divided by \(10\) is \(7\). - \(3^4 = 81\), remainder when divided by \(10\) is \(1\). - \(3^5 = 243\), remainder when divided by \(10\) is \(3\) (and the pattern starts to repeat). From this computation, we observe a repeating cycle in the remainders: \(3, 9, 7, 1\). The cycle has a length of \(4\). To determine which value in the cycle corresponds to \(3^{102}\), we need to find \(102 \mod 4\): \[ 102 \div 4 = 25\quad \text{

To find the remainder when (3^{102}) is divided by (10), we can utilize patterns in the powers of (3) modulo (10).

Calculating the first few powers of (3):

• (3^1 = 3), remainder when divided by (10) is (3).

• (3^2 = 9), remainder when divided by (10) is (9).

• (3^3 = 27), remainder when divided by (10) is (7).

• (3^4 = 81), remainder when divided by (10) is (1).

• (3^5 = 243), remainder when divided by (10) is (3) (and the pattern starts to repeat).

From this computation, we observe a repeating cycle in the remainders: (3, 9, 7, 1). The cycle has a length of (4).

To determine which value in the cycle corresponds to (3^{102}), we need to find (102 \mod 4):

[

102 \div 4 = 25\quad \text{