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Multiplication Tables Modulo N: Homework 4 for Math 170 - Prof. E. So, Assignments of Mathematics
Instructions and examples for completing homework 4 in math 170, which involves creating multiplication tables modulo n and finding multiplicative inverses. Students are expected to write out multiplication tables for various moduli and identify pairs of numbers that are multiplicative inverses.
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(Math 170) Homework 4:
Due October 9, 2008
For a natural number n a multiplication table mod n consists is a grid such that
- The rows and columns are each labeled with numbers 0 to n − 1
- The number in the ath column and the bth row is a × b mod n.
For example, the multiplication tables mod 10 is the following:
mod 10mod 10mod 10 000 111 222 333 444 555 666 777 888 999 000 0 0 0 0 0 0 0 0 0 0 111 0 1 2 3 4 5 6 7 8 9 222 0 2 4 6 8 0 2 4 6 8 333 0 3 6 9 2 5 8 1 4 7 444 0 4 8 2 6 0 4 8 2 6 555 0 5 0 5 0 5 0 5 0 5 666 0 6 2 8 4 0 6 2 8 4 777 0 7 4 1 8 5 2 9 6 3 888 0 8 6 4 2 0 8 6 4 2 999 0 9 8 7 6 5 4 3 2 1
And the multiplication tables mod 9 is the following:
mod 9mod 9mod 9 000 111 222 333 444 555 666 777 888 000 0 0 0 0 0 0 0 0 0 111 0 1 2 3 4 5 6 7 8 222 0 2 4 6 8 1 3 5 7 333 0 3 6 0 3 6 0 3 6 444 0 4 8 3 7 2 6 1 5 555 0 5 1 6 2 7 3 8 4 666 0 6 3 0 6 3 0 6 3 777 0 7 5 3 1 8 6 4 2 888 0 8 7 6 5 4 3 2 1
So given this table to find 7 × 4 mod 9 I look at the row labeled 7 and the column labeled 4 and find their intersection and I see the answer is 1.
Exercise 1: Write out a multiplication table mod 2, mod 3, mod 4, mod 5, mod 6, mod 7, mod 8.
Exercise 2: For each table in Exercise 1 and the two tables on this sheet find the pairs of numbers which are multiplicative inverses (i.e. a and b such that a × b = 1 mod n).
For which tables does every number have a multiplicative inverse? Do you think every number on the multiplicative table mod 11 will have an inverse? What about mod 12? Can you make any general predic- tions?
Exercise 3: Heart of Mathematics Chapter 2.4 Exercise 3
Exercise 4: Heart of Mathematics Chapter 2.4 Exercise 5
Exercise 5: Heart of Mathematics Chapter 2.4 Exercise 8
Exercise 6: Heart of Mathematics Chapter 2.4 Exercise 18
Exercise 7: Heart of Mathematics Chapter 2.4 Exercise 24
Exercise 8: Heart of Mathematics Chapter 2.4 Exercise 32
Exercise 9: Heart of Mathematics Chapter 2.4 Exercise 39