Sections, concepts, and problems

2.1. Algorithms. Algorithms, pseudocode for describing them, and two examples of search algorithms: linear and binary search algorithms. Each algorithms has an input and output; the correct output should be obtained after a finite number of steps. (You might have learned your first algorithm when you learned to tie your shoes.)
1, 3, 5, 7, 9, 27

2.2. Complexity of algorithms. Time complexity of algorithms and big-O estimates of this quantity.
5, 7, 9abdef

2.3. The integers and division. Divisibility, the fundamental theorem of arithmetic, the division algorithm, greatest common divisors, least common multiples, modular arithmetic.
7, 9abg, 11, 13 (note: if it is a/b, then derive 2^a=3^b, which is impossible by the Fundamental Theorem of Arithmetic), 15b, 21, 29, 35

2.4. Integers and algorithms. The Euclidean Algorithm (for finding the gcd of 2 integers) and an algorithm for converting an integer from base 10 to another base b.
1abc, 3, 5a, 7c, 11c, 15 [hint: write a=10ⁿa_n + ... + 10 a₁ + a₀. Then since 10 is congruent to 1(mod 3), a is congruent to a_n + ... + a₁ + a₀.]

2.5. Applications of number theory. Writing the gcd of two positive integers as a Z-linear combination of the two integers, multiplicative inverses (mod m) and when they exist, linear congruences, the Chinese Remainder Theorem, Fermat's Little Theorem.
1df, 3, 7, 9, 11, 15 [hint: x²-1 factors], 23 [for (c) you need problem 19]

2.6. Matrices. Addition and multiplication of matrices, the transposes, symmetric matrices, diagonal matrices, the identity matrix, inverses, zero-one matrices and their join, meet, and Boolean product.
1, 3c, 11, 14, 15, 27, 29

Other Chapters

Chapter 1 | Chapter 2 | Chapter 3 | Chapter 4 | Chapter 6