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Name
INTRODUCTION TO NUMBER THEORY
Exam 3
April 28, 2000
The point value of each problem is given in the margin.
(10) 1. Find the least complete solution of the congruence
15 x ≡ 100 (mod 35).
(20) 2. Short answer.
a) Give a reduced residue system modulo 20.
b) Let p be a prime. How many solutions (mod p^3 ) does the congruence px ≡ b (mod p^3 ) have if p|b?
How about if p - b?
c) Evaluate
19
d) If m is an odd number such that 2m−^1 ≡ − 1 (mod m) can we make any conclusion as to whether m is a prime or not? Explain.
(12) 3. Find an integer x with 100 < x < 200 such that 4x ≡ 1 (mod 11), and x ≡ 2 (mod 9).
(10) 4. Say the decimal expansion of 3/140 is given by
3 140
= .a 1 a 2... aic 1 c 2... ck
with i, k minimal. Find the values of i, k.
(10) 5. a) What is the order of 4 (mod 11)?
b) Find the remainder in dividing 4^46 by eleven.
(14) 8. In the RSA method of public cryptography suppose that you have chosen p = 5, q = 11, e = 7. a) Calculate the least common multiple [p − 1 , q − 1] =
b) Calculate the decode exponent d.
c) Suppose that someone has sent you the encrypted message M 1 , that is M 1 ≡ M 0 e (mod pq), 0 < M 1 < pq, and M 0 is the original message, 0 < M 0 < pq. Explain how you will decipher the message.
d) Which quantities are made public?
e) In practice, the primes chosen are much larger. What is it that allows the RSA method to be secure and yet be public?
