Final answer:
Using Euler's theorem to find 2¹ Mod 22, we first determine that 2 and 22 are coprime. Simplifying the exponent, we use 2¹³ Mod 22 to find the answer, which is 8. The correct choice is not listed in the provided options.
Step-by-step explanation:
To calculate 2¹ Mod 22, we can use Euler's theorem, which states that if two numbers are coprime, the number a raised to the power of the Euler's totient function of n will be equivalent to 1 modulo n.
In this case, the numbers 2 and 22 are coprime (they have no common divisors other than 1). The Euler's totient function of 22 (φ(22)) is 10, as 22 has 1/11 prime factors and φ(p^k)=p^k - p^(k-1) for prime p and k ≥ 1. We can simplify our problem by writing the exponent 53 as a multiple of 10 plus a remainder, so 53 = 5*10 + 3. Thus 2¹³ = (2¹°)¹*2¹³. According to Euler's theorem, (2¹°)¹ = 1 since 10 is φ(22), therefore we only have to calculate the value of 2¹³ mod 22. As 2¹³ = 8, which is smaller than 22 we conclude 2¹³ mod 22 = 8. The correct answer to this question is not amongst the available choices given (A) 1, (B) 10, (C) 2, (D) 0; therefore the student should be advised to review the options provided or ensure there are no typos in the question.