1. Z/nZ

Can anyone please explain to me how the "Z/nZ" thing work? any examples?

my professor mentioned it several times in class, for it is not on the textbook.

thanks so much

Can anyone please explain to me how the "Z/nZ" thing work? any examples?

my professor mentioned it several times in class, for it is not on the textbook.

thanks so much
Z/nZ is the set of all integers "modulo n". Two numbers are "congruent mod n" if and only if the give the same remainder when divided by n. For example if n= 3, then all multiples of 3 give the same remainder, 0, when divided by 3. They are all congruent modulo 3. On the other hand, the numbers 1, -2, 4, -5, 7, -8, etc., all numbers of the form 3n+1 for some integer n, have remainder 1 when divided by 3 so are all congruent mod 3. Finally, the numbers of the form 3n+ 2 all have remainder 2 when divided by 3. Since the remainder, on dividing by 3, must be 0, 1, or 2, those are the only 3 possiblities.

We make three sets of these {3k}, that is the set of all multiples of 3, are in one set, all numbers that have remainder 1 when divided by 3, {3k+1}, are in another, and all numbers that have remainder 2 when divide by 3, {3k+2} are in a third.

And now we make define addition and multiplication of those sets! For example, to add {3k+1} and {3k+ 2}, select a "representative" of each set, one number out of each set. For example 28= 3(9)+ 1 so 28 is in the first set. 14= 3(4)+ 2 is in the second set. Adding those numbers, 28+ 14= 42 and 42= 3(14) so is in {3k}: {3k+1}+ {3k+2}= {3k}. Suppose we had chosen instead 7= 3(2)+1 from the first set and 8= 3(2)+ 2 from the second- then 7+ 8= 15= 3(5) so we still get a multiple of 3: {3k}. We can show that in general. Any member of {3k+1} is of the form 3m+1 for m some integer and any member of {3k+2} is of the form 3n+ 2 (I have changed from the general "k" to m and n since the particular multiplier doesn't have to be the same). Then 3m+1+ 3n+ 2= 3m+3n+ 3= 3(m+n+1), a multiple of 3.

Since it doesn't matter what representative we choose we can represent those sets by the smallest non-negative member: (0), (1), and (2) or, if it is understood that we are working in Z/3Z, "integers modulo 3", just 0, 1, and 2.

0+ 0= 0, 0+ 1= 1, 0+ 2= 2, 1+0= 1, 1+ 1= 2, 1+ 2= 0 (because 1+ 3= 3 which is a multiple of 3 and that set is represented by 0), 2+0= 0, 2+ 1= 0, 2+ 2= 1 (because 2+ 2= 4= 3+1). That is, add the two numbers and then represent the sum by the smallest non-negative integer that is congruent to the sum.

Same thing with multiplication: 2*2= 4= 3+ 1 so "2*2= 1 mod 3".