Download Test 2 Review

Math 113
Test 2 Review
Spring '17
1. True or false; briefly explain your answer.
a) The product of any two irrational numbers is always irrational.
b) There is a largest negative real number.
c) There is a smallest real number that is > 1.
d) The number x = 0.1234123412341234 . . . . in which 1234 repeats forever is rational.
e) There is a largest negative real number.
f) The irrational number 5 has a repeating (ie periodic) decimal expansion.
g) The sum of any two irrational numbers is always irrational.
h) There a smallest positive real number.
i) The number x = 0.12345678910111213 . . . . which is constructed by writing
all
of the natural numbers in order after the decimal point, is irrational.
j) The sum of any two rational numbers is always rational.
k) The irrational number 2 has a repeating (ie periodic) decimal expansion.
x
2. a) If x solves the equation 12 = 7 , prove that x is an irrational number.
b) Prove that √(11) an irrational number. Be sure to explain all of the steps.
3. a) Explain what it means to say that two infinite sets have the same cardinality.
b) What did Georg Cantor prove about the cardinality of the reals compared to the
natural numbers?
4. Billy Bob says he has found a one - to - one correspondence between the natural
numbers and the real numbers between zero and one whose decimals consist of only 0, 1,
2 and 3. “Fool! You should lay off the cheap beer at night.” says Bubba. “I’ll show you.”
says Billy Bob. “ Here are the first few numbers in my list.” Bubba, of course, has also
had a few too many cheap beers. Help him out by finding the first few decimals of a
number not on Billy Bob’s list. Explain your method so even Billy Bob can understand.
Billy Bob’s list
1
0.012113030302120221130302 . . .
2
0.022101230303130313 . . .
3
0.233202033021201131313 . . .
4
0.13330112122212013012230121110201 . . .
5
0.12130212022023 . . .
5. Show that the sets listed below have the same cardinality as the natural numbers.
N = { 1, 2, 3, 4, 5, 6, 7, 8, . . . . } Write an explicit formula that matches them.
A = { -3, 0, 3, 6, 9, 12 , . . . 3n, . . . . }
B = { -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, . . . . }
C = { 4, 7, 10, 13, 16, 19, 22, 3n + 1, . . . . }
D = { -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, . . . . }
6. Write the periodic decimal number x = 0.123123123 . . . . in which 123 repeats
forever as a ratio of two integers. Show how you are determining this.
7. Recall that for public key encryption there are 2 primes p and q. The number n = pq
and m = (p-1)(q-1) The encryption exponent e can have no common factors with m and
the decryption exponent d satisfies de = 1(mod m).
a) Which of these two numbers are made public?
b) How would a number x be encrypted?
c) How is the encrypted number decrypted?
d) Let p = 11 and q = 17. Determine n and m and circle a valid value of e from the list
below and for this choice of e determine the correct value of d (by trial and error). Show
your computations.
e = 11
17
25
27
d = 53
83
113
131
n = _______ m = ______
e = _______
d = _________
8. (10 pts.) In the 9 digit bank code shown below, the first digit is listed as x because it
was illegible. Determine x so that the code is valid.
In a bank code, each digit is multiplied by 7 3 9 7 3 9 7 3 9 and the sum of these 9
terms are added modulo 10, the result should be zero.
x 2 3
7 3 9
4 5 6
7 3 9
7 0 1
7 3 9
9. (10 pts.) In the UPC code shown below, the first digit is listed as x because it was
illegible. Determine x so that the code is valid.
In a UPC code, if each digit is multiplied by alternating 3 1 3 1 …. 3 1 and the
sum of these 12 terms are added modulo 10, the result should be zero.
x
3
2 3 4 5 6 7 0 0 0 0 4
1 3 1 3 1 3 1 3 1 3 1
10. a) 5479 is a prime number. What does Fermat's little theorem say about the value
of 25478 (modulo 5479)?
b) What is 210960 (modulo 5479) (The answer should be a number between zero and
5479.)