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In this paper, we report the experience of a mathematics teacher educator, who is a mathematician, while teaching the Euclidean Division Algorithm Theorem in a Number Theory course for prospective mathematics teachers. Considering that the knowledge of a Mathematics teacher is specialized, from the perspective of the Mathematics Teachers' Specialised Knowledge model, we intended to identify which knowledge is mobilized when the teacher educator addresses this algebraic result. Some activities were conducted by the professor in an attempt to understand how the prospective teachers performed the division of integers before, during, and after knowing the theorem; our analysis focuses on these different moments. Regarding the prospective teachers' knowledge, it was possible to observe, primarily, knowledge related to procedures involving the algorithm. However, throughout the conducted activities, they were able to establish different connections concerning the Euclidean Division Algorithm Theorem. About the teacher educator, we emphasize that his mathematical and pedagogical knowledge, combined with the goal of effectively prepare future Mathematics teachers, has the potential to promote specialized knowledge in the prospective teachers regarding the subject matter. [ABSTRACT FROM AUTHOR]