<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 Let \( T \) be the set of all positive divisors of \( 20044^{100} \). The subset \( S \) of \( T \) has the property that no element in \( S \) is an integer multiple of any other element in \( S \). Find the maximum possible number of elements in \( S \). Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
10201
Let \( T \) be the set of all positive divisors of \( 20044^{100} \). The subset \( S \) of \( T \) has the property that no element in \( S \) is an integer multiple of any other element in \( S \). Find the maximum possible number of elements in \( S \).
test