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

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The group U100, consisting of the units modulo 100, contains 40 elements. There are no elements of order 50 within this group. Instead, the discussion clarifies that elements satisfying the equation a^50 = 1 correspond to those that satisfy a^10 = 1, resulting in 20 elements that fulfill this condition. The correct approach to determine the number of elements of a specific order involves understanding the group's structure and applying the properties of cyclic groups.

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feyomi
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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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