If there are 10 chromosomes in a particular cell at the start of prophase, how many will be present in the same cell at the end of anaphase, before cytokinesis has begun? How many will there be after cytokinesis has occurred?

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Prophase, anaphase and cytokinesis are parts of mitosis. Mitosis is cellular division in which one parent cell divides into two daughter cells. It makes more copies of somatic (body) cells, as opposed to meiosis that makes sex cells. Therefore, mitosis maintains the number of chromosomes from parent to daughter cells. The steps of mitosis, in order, are prophase, metaphase, anaphase, and telophase. 

Towards the beginning of prophase, each chromosome duplicates (makes a copy) if itself. During prophase, the chromosomes condense and become visible. These chromosomes line up in the middle of the cell during metaphase. The duplicated chromosomes are pulled apart from one another during anaphase.  During telophase, the cell begins to furrow (split). Finally, the parent cell is cut into two daughter cells during cytokinesis. 

So, in regards to your question, if there are 10 chromosomes in a particular cell at the start of prophase (assuming that duplication has already occurred), then there would still be 10 chromosomes in the cell at the end of anaphase, before cytokinesis has begun. However, once cytokinesis occurs, each daughter cell would have the original number of 5 chromosomes. 

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