Finding the Remainder of $2^{20}$ Divided by 3

To find the remainder when $2^{20}$ is divided by 3, we can use modular arithmetic. We look for a pattern in the remainders of the powers of 2 when divided by 3.

Analyzing Powers of 2 Modulo 3

• $2^1 \pmod{3}$: $2 \div 3$ leaves a remainder of 2. So, $2^1 \equiv 2 \pmod{3}$.
• $2^2 \pmod{3}$: $4 \div 3$ leaves a remainder of 1. So, $2^2 \equiv 1 \pmod{3}$.
• $2^3 \pmod{3}$: $8 \div 3$ leaves a remainder of 2. So, $2^3 \equiv 2 \pmod{3}$.
• $2^4 \pmod{3}$: $16 \div 3$ leaves a remainder of 1. So, $2^4 \equiv 1 \pmod{3}$.

Identifying the Pattern

The pattern of the remainders is 2, 1, 2, 1, ...

• When the exponent ($n$) is odd, $2^n \equiv 2 \pmod{3}$.
• When the exponent ($n$) is even, $2^n \equiv 1 \pmod{3}$.

Calculating the Remainder for $2^{20}$

The exponent in the question is 20.

• Since 20 is an even number, the remainder when $2^{20}$ is divided by 3 follows the pattern for even exponents.
• Therefore, $2^{20} \equiv 1 \pmod{3}$.

The remainder is 1.