Math Foundations: Previous Test 1

No books or notes may be used. You may use a calculator to check answers and do routine calculations. However, you also must show all your work. Answer all questions in the spaces provided. For problem solving questions, be sure to indicate the major steps and the final answer.

1. (3 points) Let a, b, and c be integers and ab ≠ 0. Prove: If a | b and b | c, then a | c.
   

1. (2 points) State the division algorithm.

1. (2 point) Apply the division algorithm to determine the quotient and remainder in -95 = ____* (8) + ____.

1. (3 points) Let be positive integers. Prove or disprove:
   if and, then .
   
2. (2 points) Solve the congruence
   

1. (1 point) Suppose that a computer has only the memory locations 012…17. Use the hashing function h, where , where x is the record ID, to determine the memory locations in which the record with ID 53 is stored.

1. (2 points) Give the sequence of pseudorandom numbers generated using the pure multiplicative generator with seed ?
   
2. (2 point) Give the prime factorization of 10780.

1. (4 points) If and , what are gcd(a,b) and lcm(a,b)?
   

1. (3 points) Use the Euclidean algorithm to find gcd(68, 86).
   

1. (3 points) Find the Bezout coefficients for 68 and 86. That is, find integers s and t such that gcd(68, 86) = 68s + 86t.

1. (6 points) Give the binary, octal and hexadecimal representations of the base-10 number 2xyzw, where xy and zw represent the two-digit month and day values, respectively, for your birthday. (E.g., I was born March 2 … so my number would be 20302.)

1. (6 points) Give the binary, decimal and hexadecimal representations of .
   

1. (5 points) Let In be an nn identity matrix and Dn an nn diagonal matrix. Which of the following statements are true? Give a reason or counterexample.

Statement True/False Why? .
In is a lower triangular matrix. T F ______
Dn is a symmetric matrix. T F ______
In is a stochastic matrix. T F ______
What is the matrix specified by?

______

1. (3 points) Suppose A is a 55 matrix and B a 65 matrix. Is A+B defined? Is AB defined? Is BA defined? Give very brief justifications of your answers.

1. (2 points) Suppose. What is? ______

1. (3 points) Calculate. ______
   
2. (3 points) Let A, B and C be matrices of sizes 104, 410, 105, respectively. What is the most efficient way to calculate the product ABC --- (AB)C or A(BC)? Give the number of multiplications required for each process.
   
3. (2 points) If , find .

1. (2 points) For the matrix , find

1. (4 points) Rewrite the following system of equations as a matrix equation.

1. (4 points) Give the inverse of using the augmented matrix method (Gauss-Jordan elimination).
   

BONUS:1) Are there two-digit decimal and hexadecimal numbers (non-zero) such that ?

What about the same for two-digit decimal and octal numbers?

If so, show the number(s). If not, show why not.

2) Show the binary representation for the following approximation of π 3.14 accurate to the

place.