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

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):

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102 \div 4 = 25\quad \text{