Discussion question 2.5 & 2.6: What is a multiplicative inverse? Is Z_8 cyclic? What about U_8? 1) Does 2 have a multiplicative inverse in Z_8? i.e. does the equation 2x=1 have a solution? Check every possibility: PyhJImFHNiIiIiEiIiIiIihJJXRydWVHJSpwcm90ZWN0ZWRHLUkmcHJpbnRHRik2JEYjLUkkbW9kR0YkNiQsJEYjIiIjIiIp print(); # input placeholder We find that U_8 = {1,3,5,7}. 2) Is Z_8 cyclic? Is g=3 a generator? Check:JSFH PyhJImFHNiIiIiEiIiIiIihJJXRydWVHJSpwcm90ZWN0ZWRHLUkmcHJpbnRHRik2JEYjLUkkbW9kR0YkNiQsJEYjIiIkIiIp print(); # input placeholder Since {0,3,6,1,4,7,2,5} = {0,1,2,3,4,5,6,7}, we see that g=3 is a generator of Z_8. 3) Is U_8 cyclic? Is g=3 a generator? Check: PyhJImFHNiIiIiEiIiIiIzVJJXRydWVHJSpwcm90ZWN0ZWRHLUkmcHJpbnRHRik2JEYjLUkkbW9kR0YkNiQpIiIkRiMiIik= print(); # input placeholder Conclusion? PyZJImdHNiI8JiIiIiIiJCIiJiIiKEkldHJ1ZUclKnByb3RlY3RlZEdDJC1JJnByaW50R0YrNiNGIz8oSSJhR0YkRiZGJiIiJUYqLUYuNiRGMS1JJG1vZEdGJDYkKUYjRjEiIik= print(); # input placeholder Conclusion?