Integers Mod n Zn and Zp
1. in Z11 which numbers have square roots? What are they?
2. In Zp 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
a) show that 211872946 passes the criteria
b) does my own check routing # of 011000138 ?
c) examine your own routing number. Just report whether it passed or not.
10. What does the symbol a-2 in Zn mean, in your opinion?
