MATH 470.200/501
Examination 1
September 29, 2011
NAME
SIGNATURE
This exam consists of 8 problems, numbered 1–8. For partial credit you must present your
work clearly and understandably and justify your answers.
The use of calculators is permitted on this exam.
The point value for each question is shown next to each question.
You must turn your Exam Note Sheet in with this exam.
CHECK THIS EXAMINATION BOOKLET BEFORE
YOU START. THERE SHOULD BE 8 PROBLEMS ON
7 PAGES (INCLUDING THIS ONE).
Do not mark in the box below.
Points
Possible Credit
1
16
2
24
3
16
4
8
5
20
6
15
7
8
8
12
Notes
10
Total
129
NAME
1.
MATH 470
Examination 1
Page 2
[16 points] Complete the statements of the following theorems and definitions we have
covered.
(a) Let a and m be integers. Then a has a multiplicative inverse modulo m
if and only if a and m satisfy
.
(b) (Fundamental Theorem of Arithmetic) Every integer greater than 1 can be factored into the
product of
in a
up to
way
.
(c) (Euler’s Theorem) Let m be a positive integer and let b be an integer with
gcd(m, b) = 1. If we set
φ(m) :=
,
then...
(d) (Chinese Remainder Theorem) For m and n relatively prime positive integers and
integers a and b, the congruences
x ≡ a (mod m),
x ≡ b (mod n),
can be...
September 29, 2011
NAME
2.
MATH 470
Examination 1
Page 3
[24 points; (a) 16 points; (b) 8 points]
(a) Use the Euclidean Algorithm to find gcd(2537, 4189) and to find one solution of
2537x + 4189y = gcd(2537, 4189) with x, y ∈ Z. Show your work.
(b) Using your findings in part (a), find a multiplicative inverse of 43 modulo 71.
Show your work.
September 29, 2011
NAME
3.
MATH 470
Examination 1
Page 4
[16 points] Find all congruence classes of solutions modulo 91 of the following congruences. Be sure to show your work. Solving by trial and error will yield no more
than half credit.
(a) 3x ≡ 20 (mod 91)
(b) 7x ≡ 56 (mod 91)
4.
[8 points] Calculate φ(360), where φ is Euler’s φ-function. Show your work.
September 29, 2011
NAME
5.
MATH 470
Examination 1
Page 5
[20 points; (a) & (b) 8 points; (c) 4 points] We use the following alphabet:
A B C D E F G
00 01 02 03 04 05 06
P Q R S T U V
15 16 17 18 19 20 21
H I
J K L M
07 08 09 10 11 12
W X Y Z
. ♥
22 23 24 25 26 27
N O
13 14
♣
28
(a) Using the affine cipher x → 6x + 8 (mod 29), encrypt the plaintext message
EULER into its ciphertext.
(b) Someone has sent you the message A♥♣D using the encryption key in part (a).
What did they say?
(c) Suppose we had instead used a 30 letter alphabet, and used the encryption method
x → 6x + 13 (mod 30). What problem would we have with this encryption scheme?
September 29, 2011
NAME
6.
MATH 470
Examination 1
Page 6
[15 points] Answer Yes or No to each of the following questions. In this problem
‘FLT’ stands for ‘Fermat’s Little Theorem’. Be careful: a lot of the numbers below
look alike.
(a) The number 31405529 is prime. Does FLT imply that 2001131405529 ≡ 20011
(mod 31405529)?
(b) We know that 2001131405500 ≡ 1 (mod 31405501). Does FLT imply that 31405501
is prime?
(c) We know that 2001131405480 ≡ 11935342 (mod 31405481). Does FLT imply that
31405481 is composite?
7.
[8 points] Find a primitive root modulo 7. Justify your answer.
September 29, 2011
NAME
8.
MATH 470
Examination 1
Page 7
[12 points; (a) 4 points; (b) 8 points] Suppose there is a language that only has the
letters Y and Z. The frequency of the letter Y is 0.9, and the frequency of the letter
Z is 0.1. A message is encrypted using a Vigenère cipher (working mod 2 instead of
mod 26). The ciphertext is YZYZYZZZYZ.
(a) Show that the key length is probably 2.
(b) Using the information on the frequencies of the letters, determine the key and
decrypt the message.
September 29, 2011