Finding the Remainder of $3^{256}$ divided by 5

To find the remainder when $3^{256}$ is divided by 5, we can examine the pattern of the remainders of powers of 3 when divided by 5.

• Calculate the first few powers of 3 modulo 5:
  • $3^1 \pmod{5} \equiv 3$
  • $3^2 \pmod{5} \equiv 9 \pmod{5} \equiv 4$
  • $3^3 \pmod{5} \equiv 3 \times 3^2 \pmod{5} \equiv 3 \times 4 \pmod{5} \equiv 12 \pmod{5} \equiv 2$
  • $3^4 \pmod{5} \equiv 3 \times 3^3 \pmod{5} \equiv 3 \times 2 \pmod{5} \equiv 6 \pmod{5} \equiv 1$
  • $3^5 \pmod{5} \equiv 3 \times 3^4 \pmod{5} \equiv 3 \times 1 \pmod{5} \equiv 3$
• The pattern of remainders is (3, 4, 2, 1), which repeats every 4 powers. The cycle length is 4.
• To find the remainder for $3^{256}$, we need to find the position in the cycle. This is determined by the remainder of the exponent (256) when divided by the cycle length (4).
• Calculate the remainder of the exponent:

  $256 \div 4 = 64$

  The remainder is 0. When the remainder is 0, it corresponds to the last element in the cycle (the 4th element in this case).

• The remainder corresponding to the 4th position in the cycle is 1.
• Therefore, the remainder when $3^{256}$ is divided by 5 is 1.

Final Answer: The final answer is 1