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# LS9-24 - S UMMARY OF 9/24 L ECTURE We started lecture by...

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S UMMARY OF 9/24 L ECTURE We started lecture by talking more about Z n , a cyclic group of order n which is generated by the element 1 + subject to the condition n 0 ( mod n ) . In other words, Z n = { 0 , 1 , 2 , . . . , n - 1 } under the binary operation ‘addition ( mod n ) ’: just like ordinary addition except that n is the same thing as 0, which we write in the form n 0 ( mod n ) . From this it follows that n + 1 1 ( mod n ) , n + 5 5 ( mod n ) , etc. A natural question is, if we can do addition ( mod n ) on Z n , how about multiplication ( mod n ) ? In other words, is Z n a group under multiplication ( mod n ) ? Well, it’s certainly closed, asso- ciative, and has an identity – namely, 1. But, not every element is an inverse... as usual, 0 isn’t. (Because 0 times anything is 0, and therefore will never give the identity.) OK, so we remove 0 from Z n . Now do we have a group under multiplication? After some playing around, we determined that the answer is, sometimes. The way we played around with

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