Download Review for Exam I Math 203-003 1. The last digit of a five

Review for Exam I
Math 203-003
1. The last digit of a five-digit identification number is the check digit. The check digit is assigned
so that the sum of all the digits is divisible by 7. Suppose a single-digit error has been made
in the number 36253, but the check digit is correct. Correct the number in as many ways as
possible.
2. Use the definition of congruence modulo m to determine if each of the following statements is
true or false.
(a) −83 ≡ 23 mod 2
(b) 44 ≡ 15 mod 6
(c) 32 ≡ −13 mod 15
(d) −10 ≡ −6 mod 4
3. Find the smallest positive integer b to make the congruence true.
(a) b ≡ 54 mod 16
(b) b ≡ −93 mod 7
(c) b ≡ 4522 mod 3
4. Use properties of exponents to determine 5200 mod 6.
5. In Canada, each person is assigned a Social Insurance Number (SIN). It is a nine-digit number
and the ninth digit is the check digit. The check digit uses a weighted mod 10 scheme. To
calculate the check digit, multiply each digit in an even-numbered position by 2 and add those
products, then add that sum to the sum of the digits in odd-numbered positions. The check
digit is the digit required to make the weighted sum congruent to 0 mod 10.
(a) Is 245801940 a valid SIN?
(b) Find the check digit if the first eight digits of a certain SIN are 90186435.
(c) The third digit of a certain SIN is difficult to read. Find the missing digit in the number
12?637094.
6. Determine the number represented by the following Postnet code.
7. The following ZIP+4 Postnet code contains an error. Find and correct the error.
1
8. (a) Find the sum of the measures of the vertex angles of any 9-gon.
(b) Find the measure of a vertex angle in a regular 9-gon.
9. (a) Is a triangle with side lengths of 47 inches, 85 inches and 97 inches a right triangle? Why
or why not?
(b) A flagpole is 40 feet high and casts a shadow that is 30 feet long. How far is the top of
the flagpole from the end of the shadow?
10. Find the symmetries in the following strip patterns.
(a)
(b)
11. The first 10 numbers in a sequence are as follows:
3, 2, 1, 7, −2, 11, 1, 6, 17, −9
(a) If the nth number of this sequence is denoted an , give the values of a2 , a5 and a10 .
(b) If the recursion rule for this sequence is
an = 2an−3 + an−2 − an−1 ,
list the next three numbers in this sequence.
12. Suppose an animal produces a female baby each year beginning at the start of its fifth year,
at which time it is considered an adult. Find the total number of adults and babies for the
first 10 years if we begin with one female in its first year.
13. Consider the graph in Figure 1.
(a) List each vertex and its degree.
(b) Does this graph have an Euler circuit? An Euler path? If so, give one.
(c) Does this graph have a Hamiltonian circuit? If so, give one.
14. Use Kruskal’s algorithm to find a minimal cost spanning tree in the graph in Figure 2.
2
J
H
K
L
N
M
Figure 1:
26
B
20
A
11
D
30
14
F
15
11.5
23
12
10
C
29
E
G
13
16
17
H
Figure 2:
15. The following table gives distances in miles between cities in Alabama.
Cities
Birmingham
Gadsden
Anniston
Selma
Montgomery
Birmingham
–
66
62
160
91
Gadsden
66
–
27
224
127
Anniston
62
27
–
220
110
Selma
160
224
220
–
49
Montgomery
91
127
110
49
–
(a) Draw a graph corresponding to the table.
(b) Use the nearest-neighbor algorithm starting in Birmingham to find a Hamiltonian circuit.
(c) Use the cheapest-link algorithm to find a Hamiltonian circuit.
(d) Can you find a better Hamiltonian circuit?
3
16. Consider the following graph:
A
B
(a) Duplicate edges in the graph so that the new graph has an Euler circuit. Try to use as
few new edges as possible.
(b) Duplicate edges in the graph so that the new graph has an Euler path from A to B.
Again, duplicate as few edges as possible.
4