When 2^256 is divided by 17 the remainder would be

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Modular Exponentiation Concept: We will use the concept of modular exponentiation to find the remainder when 2^256 is divided by 17. This concept states that if we want to calculate a^b mod m, we can break this down into smaller powers of a and repeatedly apply the modulus to keep the numbers small.Finding Remainder for 2^256: First, we find the remainder when 2 is raised to a power and divided by 17. We start with small powers of 2. 2^1 mod 17 = 2 2^2 mod 17 = 4 2^4 mod 17 can be found by squaring the result of 2^2 mod 17. (2^2 mod 17)^2 = 4^2 = 16 mod 17 = 16Calculating Small Powers of 2: We continue by finding 2^8 mod 17, which is the square of 2^4 mod 17. (2^4 mod 17)^2 = 16^2 mod 17 To calculate this, we can first find 16^2 = 256 and then take this modulo 17. 256 mod 17 = 0 (since 256 is a multiple of 17)Finding 2^8 mod 17: Knowing that 2^8 mod 17 = 0, we can find 2^16 mod 17 by squaring the result of 2^8 mod 17. (2^8 mod 17)^2 = 0^2 = 0 mod 17Finding 2^16 mod 17: Now, we can express 2^256 as (2^16)^16. Since we know that 2^16 mod 17 = 0, raising this to any power will still result in 0 mod 17. (2^16 mod 17)^16 = 0^16 = 0 mod 17Correcting Previous Error: However, we made a mistake in the previous steps. 2^8 mod 17 is not 0, and therefore, 2^16 mod 17 is not 0 either. We need to correct this error and re-calculate the correct remainders.

When $2^{256}$ is divided by $17$ the remainder would be

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When $2^{256}$ is divided by $17$ the remainder would be