How many elements of order 50 are there in this group?

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The group in question is U100, the group of units modulo 100, which, correct me if I'm wrong, is equal to {3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99}.

How many elements are there of order 50? How would I go about working this out, without having to explicitly calculate the order of each element?

Thanks.
 
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You are missing one element, 1. The group has order 40, so there are no elements of order 50. Maybe you ask about elements a such that a^50 = 1, or equivalently a^10 = 1, which looks like being 20 elements, half the group.
 

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