Math 5, Winter 2000

2

Midterm Problem-Solving Exam

(8 points)

5.

Here is the truth table for statements pand qfilled in with all 16
possibilities for the truth assignments in the four rows.

p

q

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

T
T
F
F

T
F
T
F

T
T
T
T

T
F
T
T

T
T
F
T

T
T
T
F

F
T
T
T

F
F
T
T

F
T
F
T

F
T
T
F

T
F
F
T

T
F
T
F

T
T
F
F

T
F
F
F

F
F
F
T

F
T
F
F

F
F
T
F

F
F
F
F

Fill in the head of each column with a statement that produces that
column. Use only the symbols p, q, ‡, ∧, ∨, ~, ( ). That is, "if…then,"
"and," "or," "negation," and "parentheses." Note that there are 16
columns in addition to the columns for pand q. On the paper you turn in,
you need not copy the table. Instead, number your answers 1 – 16.

(6 points)

6.

Let nbe a positive integer.

(a)
(b)

Show that (2n+ 1, 2n2 + 2n, 2n2 + 2n+ 1) is a Pythagorean Triple.
Find the u and v that come from Euclid's algorithm for generating
Pythagorean Triples.

(5 points)

7.

We all know that (a-b)(a+b)=a2-b2, which means that a-b
divides a2-b2, in fact, a+btimes. Moreover, a-bdivides
a3-b3,a4-b4,L, as the following equations show:
(a-b)(a2+ab+b2)=a3-b3
(a-b)(a3+a2b+ab2+b3)=a4-b4
a-b

a

4

+a3b

+a2b2

+ab

3

+b

4

=a

5

-b

5

(a)
(b)
(c)
(d)

Now, letting a = 10 and b = 1, rewrite all of the above formulas.
Explain why the foregoing shows that 9 always divides 10k - 1.
Explain why every power of 10, mod 9, is

.
Explain why 9divides a natural number nif and only if 9divides
the sum of the digits of n.
Explain why

(e)

divides a natural number nif and only if
the sum of the digits of n.

divides