Name:
Exam 1
Instructions. Answer each of the questions on your own paper, and be sure to show your work
so that partial credit can be adequately assessed. Put your name on each page of your paper.
1. [12 Points] Let 𝐴 be a set with 8 elements, 𝐵 a set with 6 elements, and 𝐶 = {𝑎, 𝑏, 𝑐, 𝑑}.
Give the number of elements in each of the following sets.
(a) The cartesian product 𝐴 × 𝐶.
(b) The power set 𝒫(𝐵). Recall that 𝒫(𝑋) is the set of all subsets of 𝑋.
(c) The set 𝒫3 (𝐴) consisting of all 3-element subsets of 𝐴.
(d) The set of all injective functions 𝑓 : 𝐶 → 𝐴.
2. [16 Points] For each positive integer 𝑛 ≥ 1, let 𝑆(𝑛) be the sum defined by
𝑆(𝑛) = 1 + 5 + 9 + ⋅ ⋅ ⋅ + (4𝑛 − 3),
which can be expressed in summation notation as
𝑛
∑
𝑆(𝑛) =
(4𝑗 − 3).
𝑗=1
(a) Compute 𝑆(1) and 𝑆(2).
(b) Use induction to prove that for every integer 𝑛 ≥ 1, 𝑆(𝑛) = 𝑛(2𝑛 − 1).
3. [16 Points]
(a) Compute the greatest common divisor 𝑑 = (1769, 2378) of the integers 1769 and 2378,
and write 𝑑 in the form 𝑑 = 1769 ⋅ 𝑠 + 2378 ⋅ 𝑡.
(b) Compute the least common multiple 𝑚 = [1769, 2378].
4. [20 Points] This exercise makes use of the following equation:
1 = 13 ⋅ 101 − 41 ⋅ 32.
Using this equation (i.e., it is not necessary to use the Euclidean algorithm to recreate it),
answer the following questions.
(a) Compute the multiplicative inverse of 32 in ℤ101 . Express your answer in the form [𝑏]101
where 0 ≤ 𝑏 < 101.
(b) Solve the congruence equation 32𝑥 ≡ 7 mod 101. Express your answer in the form [𝑏]101
where 0 ≤ 𝑏 < 101.
(c) Find all solutions to the following system of simultaneous linear congruences and give
the smallest positive solution.
Math 4023
𝑥 ≡ 8
mod 101
𝑥 ≡ 3
mod 41.
June 18, 2012
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Name:
Exam 1
5. [18 Points] This problem concerns arithmetic modulo 21. All answers should be expressed
in the form [𝑎]21 with 𝑎 an integer satisfying 0 ≤ 𝑎 < 21.
(a) Compute [9]21 + [16]221 .
(b) Compute [9]21 [16]21 + [3]21 [14]21 .
(c) Compute [5]−1
21 .
(d) List the invertible elements of ℤ21 .
(e) List the zero divisors of ℤ21 .
6. [18 Points]
(a) State Euler’s Theorem precisely. Be sure to carefully state the requisite hypotheses.
(b) Compute 𝜑(324). (Note that 324 = 81 ⋅ 4.) As usual 𝜑(𝑛) denotes the Euler 𝜑-function
applied to 𝑛.
(c) Compute 7115 mod 324.
Math 4023
June 18, 2012
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