2024 AMC 10B Problem 12

A group of 100 students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students A and B, student A speaks some language that student B does not speak, and student B speaks some language that student A does not speak. What is the least possible total number of languages spoken by all the students?

\textbf{(A)} 9 \qquad \textbf{(B)} 10 \qquad \textbf{(C)} 12 \qquad \textbf{(D)} 51 \qquad \textbf{(E)} 100

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.