$$\text{Hallar el minimo } \textit{n} \in \mathbb{N} \text{ tal que } 260 | 2^{100} + \textit{n}.$$
Ok, lo ataco por congruencia:
$$260 | 2^{100} + n \Longrightarrow n \equiv -2^{100} (260)$$
$$n \equiv -2^{100} (260) \Longleftrightarrow n \equiv -2^{100} (4) \wedge n \equiv -2^{100} (5) \wedge n \equiv -2^{100} (13)$$
$$* n \equiv -2^{100} (4) \Longleftrightarrow n \equiv 4^{50} (4) \Longleftrightarrow n \equiv 0 (4)$$
$$* n \equiv -2^{100} (5)$$
Por fermat:
<div class="math math-display">$$2^{4} \equiv 1 (5) \Longrightarrow n \equiv -{2^{4}}^25 (5) \Longleftrightarrow n \equiv -1 (5)$$</div>
$$* n \equiv -2^{100} (13)$$
Por fermat:
<div class="math math-display">$$2^{12} \equiv 1 (13) \Longrightarrow -n \equiv {2^{12}}^8 \cdot 2^{4} (13) \Longleftrightarrow -n \equiv 16 (13)$$</div>
$$\Longleftrightarrow n \equiv -3 (13) \Longleftrightarrow n \equiv 10 (13)$$
Entonces, hago el teorema chino con:
<div class="math math-display">$$\bullet n \equiv 10 (13)$$</div>
<div class="math math-display">$$\bullet n \equiv 4 (5)$$</div>
<div class="math math-display">$$\bullet n \equiv 0 (4)$$</div>
y eso da:
<div class="math math-display">$$n \equiv 244 (260)$$</div>
Y ese es el minimo n = 244