**Chapter 10**

**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*.

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