Question

What is the remainder if 2^93 is divided by 9?​

Answer 1

Answer:

Remainder is 4.

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Answer 2

Answer: The remainder when 2^93 is divided by 9 is 8.

Step-by-step explanation:

To find the remainder when 2^93 is divided by 9, we can use the concept of modular arithmetic:

1. First, we note that 9 is a factor of 10 - 1, so we can use Fermat's Little Theorem to simplify the calculation. Fermat's Little Theorem states that if p is a prime number and a is any integer not divisible by p, then a^(p-1) ≡ 1 (mod p).
2. Since 9 is not a prime number, we need to use a modified version of Fermat's Little Theorem that applies to certain composite numbers. Specifically, if a is any integer not divisible by 3, then a^6 ≡ 1 (mod 9).
3. We can use this modified theorem to find the remainder when 2^93 is divided by 9. First, we note that 2 is not divisible by 3. Then, we rewrite 2^93 as (2^6)^15 * 2^3. By the modified theorem, 2^6 ≡ 1 (mod 9), so (2^6)^15 ≡ 1^15 ≡ 1 (mod 9). Therefore, 2^93 ≡ 1 * 2^3 ≡ 8 (mod 9).

Finally, we conclude that the remainder when 2^93 is divided by 9 is 8.

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