Exercise 1. This software determines the homomorphisms from $$Z_m$$ to $$Z_n$$. (Recall that a homomorphism from $$Z_m$$ is completely determined by the image of 1.) Run the program for $$m = 20$$ with various choices for $$n$$. Run the program for $$m = 15$$ with various choices for $$n$$. What relationship do you see between $$m$$ and $$n$$ and the number of homomorphisms from $$Z_m$$ to $$Z_n$$? For each choice of $$m$$ and $$n$$, observe the smallest positive image of 1. Try to see the relationship between this image and the values of $$m$$ and $$n$$. What relationship do you see between the smallest positive image of 1 and the other images of 1? Test your conclusions with other choices of $$m$$ and $$n$$.

Please enter $$m$$ and $$n$$.