Final answer:
To find 3⁶⁴⁴ mod 645, we can use Algorithm 5. We use repeated squaring to simplify the calculation and find that 3⁶⁴⁴ mod 645 is equal to 491.
Step-by-step explanation:
To find 3⁶⁴⁴ mod 645 using Algorithm 5, we start by cubing the digit term, which is 3.
This gives us 3³ = 27.
Next, we multiply the exponent of the exponential term by 3 and reduce it modulo (φ(645) - 1), where φ is the Euler's totient function.
In this case, φ(645) = 320, so (3⁶⁴⁴ mod 645) ≡ 27^(3*(644 mod 320)) mod 645.
Evaluating the exponent, we have 3*(644 mod 320) = 3*4 = 12.
Therefore, (3⁶⁴⁴ mod 645) ≡ 27^12 mod 645.
Now, we can use repeated squaring to calculate
27^12: 27^12 = (27^6)^2 = (387420489^2) mod 645.
Simplifying further, we have
(387420489^2) mod 645 ≡ 214^2 mod 645 ≡ 45796 mod 645 = 491.
Therefore, 3⁶⁴⁴ mod 645 = 491.