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MATH 201
TEST ii
(in class)
25 October 2016
Name: _______________________
NOTE: Solutions based upon use of calculators will not earn credit.
As Gregor Samsa awoke one morning from uneasy dreams he found himself transformed in his
bed into a gigantic insect.
- Franz Kafka, The Metamorphosis
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PART I [5 pts each]: Complete each sentence below:
1.
Let p and q be integers and let m be an integer greater than 1. Then we write p ≡ q (mod m) if
_______________________________________________
2.
An equivalence relation on a set S 𝑖𝑠 a binary operation, R, that has the properties:
(i)
𝑎R𝑎 for all 𝑎 ∈ 𝑆
(ii)
__________________________
(iii)
________________________________
(Do not introduce more technical terms such as “reflexive”.)
3.
The basic version of the pigeon-hole principle states that if there are n-pigeon holes, k pigeons and ________ then
____________________________________________.
4.
The Fibonacci sequence is a recursive sequence, {F(m)}, defined by F(1) = 1, F(2) = 1, and
___________________________________________________________.
5.
Let a and b be non-zero integers. Then we say write a|b if _________________________.
6.
Let n and d be integers with d ≠ 0. The division algorithm states that there exist unique integers q and r such that
___________________________ where _______________________.
7.
The fundamental theorem of arithmetic states that if n ≥ 2 is an integer then n can be ________________________________.
8.
The Archimedean property of real numbers states that if x is any real number then ____________________________.
9.
The two basic steps In a proof by induction are called:
(A) __________________________________
(B) ___________________________________
10.
1599999 (mod 16) is congruent to _______________
11.
If a ≡ 9 (mod 11) and b ≡ 7 (mod 11), then 3a + 5b ≡ _____________ (mod 11)
12. Let S be a set. Then an equivalence relation on S corresponds to a ____________ on S and vice-versa.
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PART II
1.
[5 pts] The Lucas sequence, L(n), is defined recursively as follows:
L(1) = 2; L(2) = 1;
L(n+1) = L(n) + L(n-1) for n  2.
Compute the first 8 Lucas numbers, i.e. L(1), L(2), ...., L(8)
2. [5 pts] Explain why every integer can be expressed in the form 5n, 5n+1, 5n+2, 5n+3 or 5n+4.
3.
[5 pts] Find the units digit of 17902.
(Hint: Think mod 10.)
4. [9 pts] Define the following binary relation R on Z+ :
answer!)
(a) Is R reflexive? Why?
(b) Is R symmetric? Why?
(c) Is R transitive? Why?
For c, d  Z+, cRd if |c – d | < 5. (Justify each
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5.
[9 pts] Define the binary relation R on a non-empty set, S, of books as follows:
For a, b  S, aRb if book a costs more and contains fewer pages than book b. (Justify each answer!)
(a) Is R reflexive? Why?
(b) Is R symmetric? Why?
(c) Is R transitive? Why?
6.
[9 pts] Again, let S be a non-empty set of books. Let H be the binary relation defined by:
for a, b  S, aHb if book a costs more or contains fewer pages than book b. (Justify each answer!)
(a) Is H reflexive? Why?
(b) Is H symmetric? Why?
(c) Is H transitive? Why?
7. [9 pts]
There are 800,000 pine trees in a forest. Each pine tree has no more than 600,000 needles.
Prove that at least two trees have the same number of needles.
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8. [9 pts] Harvey Swick Middle School has 1,837 pupils currently registered.
Prove that at least 6 of the students celebrate their birthdays on the same day of the year.
PART III
Instructions: Select any 5 of the following 8 problems. You may answer more than 5 to
earn extra credit.) [10 pts each]
1.
Let a, b, c, d and m be integers, with m ≥ 2. Prove that if a ≡ b (mod m) and c ≡ d (mod m)
then ac ≡ bd (mod m).
2. We define recursively the sequence of numbers {a(0), a(1), a(2), …) as follows:
a(n) = 1 if 0 ≤ n ≤3 and
a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) for n ≥ 4.
Using induction, prove that a(n) ≡ 1 (mod 3) for all n ≥ 0.
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3. Find 1! + 2! + 3! + … + 12345! (mod 12)
4.
Let a and c be positive integers and let m ≥ 2 be an integer.
Demonstrate, by means of a counter-
example, that if ca ≡ cb (mod m) then it needn’t follow that a ≡ b (mod m).
5.
Let a, b and c be positive integers. Prove that if ca|cb then a|b.
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6.
Find the flaw in the following “proof”:
(from: A. J. Hildebrand, notes from University of Illinois)
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7. Find the flaw in the following “proof”:
(from: A. J. Hildebrand, notes from University of Illinois)
8.
Prove that 111333 + 333111 is divisible by 7.