Math 311W: Discrete Math
Homework 5, due on Friday, Feb. 17
Note: Whenever applicable, please give your final answers using the standard representatives, i.e., in the form [a]n where 0 6 a < n.
1. Solve the simultaneous linear congruence


x ≡ 6 mod 9
3x ≡ 2 mod 4


2x ≡ 4 mod 10 .
2. Recall that φ(n) is the number of invertible elements in Zn .
a) Use Theorem 1.6.5 and Theorem 1.6.6 to find φ(28).
b) Does 3 have finite multiplicative order modulo 28? Why? Find the order
of [3]28 .
c) Use (b) to find [3311 ]28 .
3. Follow the steps below to encrypt a message with a public key consisting of the
base 221 and the exponent 5. You may use a calculator to help you.
a) Use the top secret—the prime factorization 221 = 13 × 17—to find φ(221).
b) Use the Euclidean algorithm to find [5]−1
φ(221) .
c) Pick an integer from 0 to 10 to be the number of points you want to get for
this homework. Let m be that number plus 1 and write it down. Example:
if you want to get 10 points, write down “m = 11”.
d) Use the RSA encryption system to encode m with the public key (namely,
the base 221 and the exponent 5).
e) Briefly explain how the receiver can decode this message.
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