1. m mod n will have values ranging from 0 to n-1.
T or F
2. -39 MOD 5 and -38 MOD 6 are congruent.
T or F
3. The base system of the value 467 must be either decimal or octal.
T or F
4. The value 367 is a prime number.
T or F
5. The value 527 has a maximum of 22 possible prime factors because the square root of 527 is 22+.
T or F
6. A Permutation of the elements of a set is an ordered arrangement of the elements of the set.
T or F
7. P(7,3) = 210
T or F
8. C(8,5) = 56
T or F
9. Consider the following relations on {1, 2, 3 } :
R1 = { (1,1), (2,2), (3,3) }, and
R2 = { (1,2), (1,3), (2,3) }.
R1 is symmetrical and R2 is transitive
T or F
10. Using members of the set {1, 3, 4, 5, 7, 8}, the next larger P(6,3) permutation after 453 is 454.
T or F
11. The Sum Rule is applied when the tasks to be performed are disjoint.
T or F
12. According to the Pigeonhole principle, when (m+4) items are to be placed in (m+9) boxes, there will be more than one item in at least one box.
T or F
13. Pascal's Triangle yields the value of the coefficients of an algebraic expansion.
T or F
14. The probability of picking a "face" card (Jack, Queen or King) from a standard deck of playing cards is C(52,12).
T or F
15. P(n,r) is equal to or greater than C(n,r) when n => 1.
T or F
16. There are 139 positive integers not exceeding 634 that are divisible by either 7 or 11.
T or F
17. The relation An = an-1+ bn-2+ n + 2 is a linear, homogeneous relation of degree 2.
T or F
18. A brand of shirt comes in four basic colors, has male, female and unisex versions and has six sizes for each. This brand has a maximum of 13 different varieties.
T or F
PART B
1. Determine:
A). -53 MOD 7
B). -74 MOD 8
2. Determine the Base10 expansion of (D4B) Base16
3. Define if the each set of integers are mutually relatively prime. Defend your conclusion.
A). {8, 44, 55}
B). {7, 15, 26, 29, 37, 42}
4. Find the prime factors of the value 29,058. Show the result in proper exponential form.
5. Given:
A = 85
B = 553
Define by factoring:
A). gcd (A, B) show in exponential form
B). lcm (A, B) show in exponential form
6. Using the Euclidean Algorithm, determine:
GCD (3960, 3900).
7. Convert (1011 1001) Base2 to:
A). ( ) Base16
B). ( ) Base10
8. Given 2614BASE10. Determine the equivalent value in BASE3.
Hint: Use the Euclidean Algorithm
9. Define: (show intermediate work)
A. P(11,8) =
B. C(12,6) =
10. What is the coefficient of ( x^3 y^5 ) in the expansion (2x - 4y)^8 ? You may leave the answer in a proper intermediate form.
11. Each locker in a building is labeled with four upper-case alpha characters followed by three Base 16 characters. What is the maximum number of different locker numbers that can be generated?
12. A group of six fair coins are flipped eight times. What is the probability that each result has three heads in each flip?
13. f(n)= 3*f(n/2) - 6 when n is even and f(1) = -2.
a. What is the value of f(4)?
b. What is the value of f(8)?
14. How many positive integers not exceeding 6573 are divisible by neither 6 nor 15?
15. Given |A| = |B| = |C| = 60, |A INT B| = 25,
|B INT C| = 35, |A INT B INT C| = 15, and
|A UNION B UNION C| = 115 elements.
|A INT C| = ?
16. List the next SIX terms of the lexicographic ordering of the n-tuple 27436 where each digit is in the set {2,3,4,6,7}.
17. Which lottery presents the player with the best odds for winning, (A or B)? Defend your answer.
A = C(41,6)
B = C(42,5)
18. Determine if the following zero-one matrix is:
a. reflexive T or F | 0 1 1 |
b. symmetric T or F | 1 1 0 |
c. transitive T or F | 1 0 1 |
A. Develop the Basis Step of the algorithm to determine the number of terms (cardinality) of the union of n mutually intersecting sets. Show your work.
For example, the cardinality of the union of three mutually intersecting sets is C(3,1) + C(3,2) + C(3,3) = 3+3+1 = 7.
B. Determine the Base8 value of 1642Base9.