Hey, make I help you translate dis long English text to Pidgin, abi?
Understanding de Problem
We dey find way to calculate (1337)^(1337) mod 100 without using exponentiation. Dat mean say we no fit calculate (1337)^(1337) first, den take de modulus. Instead, we need find anoda way to simplify de calculation.
Reducing de Exponent
First, we go reduce de exponent (1337) to a smaller number. We fit do dis by noticing say for any n ∈ {0, 1, 2, ⋯, 99} and any r, k ∈ ℤ+, we get n^r ≡ (n + 100k)^r (mod 100). Dat property allow us to reduce de computation of (1337)^(1337) mod 100 to computing (37)^(1337) mod 100.
Applying Euler's Theorem
Next, we go apply Euler's Theorem, wey say for any a relatively prime to 100 and any r ∈ ℤ+, a^r ≡ a^[f(r)] (mod 100), where f(r) na de unique element in k ∈ {0, 1, ⋯, 39} such dat r ≡ k (mod 40). For our case, (37) dey relatively prime to 100, and f(1337) = 17, so we get (1337)^(1337) ≡ (37)^(1337) ≡ (37)^(17) (mod 100).
Handling Non-Relatively Prime Elements
We also need consider elements in {0, 1, 2, ⋯, 99} wey no dey relatively prime to 100. For elements wey be multiples of 5 but not 2, we get n^r ≡ 0 (mod 4) and n^r ≡ n*g(r) (mod 25), where g(r) na residue of r (mod φ[25] = 20). For elements wey be multiples of 2 but not 5, we get n^r ≡ 0 (mod 25) and n^r ≡ n*h(r) (mod 4), where h(r) na residue of r (mod φ[4] = 2). For elements wey be multiples of 10, we get n^r ≡ 0 (mod 100).
Computing de Value
Now wey we don reduce de exponent and apply Euler's Theorem, we fit compute de value of (37)^(17) mod 100. We fit do dis by iteratively squaring and reducing de result modulo 100. Specifically, we get:
(37)^(17) ≡ (37)^(16) * (37) (mod 100)
≡ (37^2)^8 * (37) (mod 100)
≡ (1369)^8 * (37) (mod 100)
≡ (69)^8 * (37) (mod 100)
≡ (61)^4 * (37) (mod 100)
≡ (3721)^2 * (37) (mod 100)
≡ (21)^2 * (37) (mod 100)
≡ (441) * (37) (mod 100)
≡ (41) * (37) (mod 100)
≡ 1517 (mod 100)
≡ 17 (mod 100)
So, de final answer na 17.
Self-Reflection
For dis problem, we apply various mathematical concepts, including modular arithmetic, Euler's Theorem, and exponentiation by squaring. We also break down de problem into manageable parts and use iterative squaring to compute de value of (37)^(17) mod 100. De key insight be to reduce de exponent using Euler's Theorem and den apply modular arithmetic to simplify de calculation.
Note: I break down de long English text into smaller sections and use Pidgin phrases to make it more readable and relatable. I also try to maintain de original meaning and explanation while using Pidgin vocabulary and grammatical structure.
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