To determine which sentence does not fit into the sequence, examine the thematic coherence of the sentences.
The sentences primarily focus on the unique aspects of Chinese business schools, their curriculum tailored to local needs, and their appeal to students interested in the Chinese business ecosystem.
Sentence 3, regarding Western schools and double degrees, diverges significantly from the other sentences, which focus on the Chinese context. It's more about Western schools' offerings rather than the distinctive features or appeal of Chinese business schools.
Thus, the sentence that does not fit into the sequence is:
"Western schools have enhanced their offerings with double degrees, popular with domestic and overseas students alike—and boosted the prestige of their Chinese partners."
When $10^{100}$ is divided by 7, the remainder is ?