Math 5248 Lec 2 Exam 1 Solutions
RSA: (82)(96)=7872.
We want the inverse of 1243 mod 7872.
7872=(6)(1243)+414
1243=(3)(414)+1
Then 1=1243-(3)(414)=1243-3(7872-6*1243)=19*1243-3*7872., so the Mult Inv of 1243 mod 7872 is
19. Then using the method in the bo
Math 5248 Lec 2 Homework #4 Due April 2.
As always, you must show your work.
10 POINTS EACH PROBLEM FOR A TOTAL OF 130 POINTS
9.1.5:
1
9.5.7 The MI of 23 is 23519%521 = 68. (3pts)
Why this works: Mod 521,
= 1, so 23519 is the MI of 23.(7pts)
9.6.6: 2260%1
Quadratic reciprocity and its application in the Euler test
1. Simple example: Is 21 a prime? Use b = 8.
since 215 mod 8.
On the one hand,
2. Lets try to determine if 1729 is prime or composite. This is an interesting number. Not only
is it a Carmichael n
Math 5248 Lec 2 Homework 5 Due 4/30/13 REVISED DUE DATE 5/2/13
Ten questions, ten points each. 100 points total.
Chapter 17: Let n = 1105= 51317.
1. How many numbers b are there in the range 1<b<n such that gcd(b,n) =1.
ANS: There are (1105)=(5)(13)(17)=4
Math 5248 Lec 2 Homework #2 Due 2/14/2013
2.3.1 PROB(10 no e) = (10.1167)10=(0.8833)10=0.2891 (approx).
Prob(100 no e)=(0.8833)100= 0.00000408153 (approx)
Your answer must be correct to at least two significant digits to get full credit. An answer of
zero
Math 5248 Lec 2 Homework 3 Due Thursday March 7
Five questions, 20 points each.
I. Find all square roots of 4 modulo 5711. A direct search which does not use the methods of
this course gets no credit. If you find all the roots, you get 20 points; otherwis
Math 5248-002 Homework #1 Answers
1.1.10: Lets look at the first word. If I shift forward by 1 or 2 steps, I get nonsense. But if I
shift forward by 3, I get BUT. This looks promising, so lets try a forward shift of 3:
BUT THIS EXAMPLE IS LESS EASY.
1.2.1
Announcements 1/29/2012
1. As announced in a previous e-mail, the due date for the first homework has been changed to
Tuesday Feb. 5.
2. In homework problem 1.2.19, since we are working mod 100 , the answer should be greater
than or equal to 0 and less th
Math 5248 Lec 2 Homework 5 Due 4/30/13
Ten questions, ten points each. 100 points total.
Chapter 17: Let n = 1105= 51317.
1. How many numbers b are there in the range 1<b<n such that gcd(b,n) =1.
2. Use the CRT to count how many solutions there are t o bn