Problem # 4
In multiplication tables, the last row is always a reverse of the first row. |
Since N - 1 is always a coprime with N, then according to Problem #5, the last row must be a permutation of the first one. It's a very specific permutation. The number a_{m} in column m staisfies
Problem # 5
In multiplication tables modulo N, rows corresponding to numbers coprime with N contain permutations of the first row. |
Let m be coprime to N. Let a and b be two different remainders of division by N. Then
Problem # 6
For prime (N + 1), multiplication tables offer multiple and simultaneous solutions to the rook problem:
on an |
All remainders of division by a prime
Problem # 7
Under the same conditions, 1 always appears in the upper left and lower right corners and nowhere else on the main diagonal. |
We claim that the equation
This fact is used to prove the Wilson's Theorem.
Problem # 12
If the table has an odd number of rows, then every remainder occurs on the main diagonal. |
Let n be the modulus and an odd integer. Every remainder appears exactly once in each row of the addition table and so appears exactly n times. Since the table is symmetric about the main diagonal, each remainder appears as many times above the diagonal as below it; hence it appears an even number of times off the diagonal. Since the remainder appears the odd number n times altogether, it must appear on the main diagonal. Thus every remainder appears on the main diagonal, and so appears exactly once there.
Of course, as a problem in group theory, this problem is almost trivial. In a finite group of odd order, every element has odd order. So, if x is an element in such a group, then
- Modular Arithmetic
- Differences and Similarities
- Solutions to some problems
- Chinese Remainder Theorem
- Euclid's Algorithm
- Pick's Theorem
- Fermat's Little Theorem
- Wilson's Theorem
- Euler's Function
- Divisibility Criteria
- Examples
- Equivalence relations
- A real life story
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