Complete residue system modulo m
Okay kiddo, so let's say you have some numbers like 2, 4, and 6. Now, what if I asked you to find all the numbers that leave a remainder of 1 when you divide them by 3? What would you do?
Well, you might start with 1 and add 3 each time: 1, 4, 7, 10, and so on. But wait, those are too big! You only want numbers less than or equal to 6. So, you go back and "wrap around" to get 1, 4, and 7 becomes 1. Now you have your answer: the numbers that leave a remainder of 1 when you divide them by 3 are 1 and 4.
That's called a "complete residue system modulo 3." The idea is to list all the possible remainders you can get when you divide numbers by 3, and then pick one number from each group to make a list of "representatives." The list is complete because it includes one representative from each group, and it's a residue system because every number that leaves a remainder of 0, 1, or 2 when you divide it by 3 is equivalent to exactly one representative on the list.
We can do this for any number m. To create a complete residue system modulo m, we list all the possible remainders you can get when you divide numbers by m, and then pick one number from each group to make a list of "representatives." This list is complete because it includes one representative from each group, and it's a residue system because every number that leaves a remainder of 0, 1, 2, ..., m-1 when you divide it by m is equivalent to exactly one representative on the list.
For example, a complete residue system modulo 4 is {0, 1, 2, 3}. That's because there are four possible remainders when you divide numbers by 4, and we picked one representative from each group: 0, 1, 2, and 3.
I hope that helps, kiddo! Let me know if you have any more questions.
Well, you might start with 1 and add 3 each time: 1, 4, 7, 10, and so on. But wait, those are too big! You only want numbers less than or equal to 6. So, you go back and "wrap around" to get 1, 4, and 7 becomes 1. Now you have your answer: the numbers that leave a remainder of 1 when you divide them by 3 are 1 and 4.
That's called a "complete residue system modulo 3." The idea is to list all the possible remainders you can get when you divide numbers by 3, and then pick one number from each group to make a list of "representatives." The list is complete because it includes one representative from each group, and it's a residue system because every number that leaves a remainder of 0, 1, or 2 when you divide it by 3 is equivalent to exactly one representative on the list.
We can do this for any number m. To create a complete residue system modulo m, we list all the possible remainders you can get when you divide numbers by m, and then pick one number from each group to make a list of "representatives." This list is complete because it includes one representative from each group, and it's a residue system because every number that leaves a remainder of 0, 1, 2, ..., m-1 when you divide it by m is equivalent to exactly one representative on the list.
For example, a complete residue system modulo 4 is {0, 1, 2, 3}. That's because there are four possible remainders when you divide numbers by 4, and we picked one representative from each group: 0, 1, 2, and 3.
I hope that helps, kiddo! Let me know if you have any more questions.
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