MME529 Homework #6  Solved

Description

Integers Mod n               Z_n     and   Z_p

 

1. in Z₁₁ which numbers have square roots?  What are they?
2. In Z_p   show that   x²  ≡  (p – x)²  mod p.

How does this help with square roots?

Give two examples to illustrate.

3. Solve 17x  =  5  mod 29.   Show all steps.   (by hand)
4. If we have ax  ≡ ay  mod n   can we always cancel the  a out ?  What do you think?
5. Simplify 889345234 mod 25  without  doing out the long division.
6. Predict with algebra which members of Z₁₅ will have multiplicative inverse.
7. Solve x²  -2x  + 2 = 0  mod 13.   Show all steps.  Check your answers.
8. Suppose for sake of discussion we are in Z₁₃. Show that a = 2  is a generator for Z₁₃ in the sense that: every member of Z₁₃  is a power of 2  (except 0 , of course).  For example  9 ≡ 2⁸ mod 13  (kinda wrecks your notion of even numbers, doesn’t it?) What happens if you try to use a = 5 as a generator?

Can you find another generator for Z₁₃  ?

9. A bank routing number appears in the lower left of all of your checks. Its purpose is to see the check is routed to the correct bank. It is  9 digits.

To increase the chances of detecting an error, the numbers as a group must satisfy an algebraic criteria using mod 10 arithmetic.  Specifically   if  ABCDEFGHI   is the routing number then

7A + 3B + 9C +7D + 3E + 9F + 7G + 3H + 9 I  mod 10  must be congruent to  0

1. show that 211872946  passes the criteria
2. does my own check routing # of 011000138  ?
3. examine your own routing number. Just report whether it passed or not.

 

10. What does the symbol a^-2  in  Z_n  mean, in your opinion?