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G = <a>, where |a| = 360. a. Find the number of elements of G with...

G = <a>, where |a| = 360.

a. Find the number of elements of G with the order 40.

b. Find the number of elements of G with the order 75.

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Answer #1

Elements of G are of the form

We want to find those elements for which where

We have

So we must have

Which means there are 16 such elements with order 40

And similarly which is impossible

Meaning there are 0 elements with order 75

(order of an element must divide size of group aka Lagrange's theorem produces contradiction)

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