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Proof Suffices - Mathematics - Exam, Exams of Mathematics

Aliah UniversityMathematics

This is the Exam of Mathematics which includes Solvability, Function, Initial Value Problem, Determine, Regular or Irregular, Equal Difficulty, Incorrect Answers etc. Key important points are: Proof Suffices, Statements, Holds, Counterexample, Prime, Congruences, Least Nonnegative Integer, Satisfies, Decimal Expansion, Last Three Digits

Typology: Exams

2012/2013

Uploaded on 02/21/2013

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Math 312, Section 101
Final Exam
December 5, 2008
Duration: 150 minutes
Please identify yourself in ink with name, student number and signature.
Name: Student Number: Signature:
Do not open this test until instructed to do so! This exam should have 10 pages, including this
cover sheet. No textbooks, notes, calculators, or other aids are allowed. Turn off any cell phones,
pagers, etc. that could make noise during the exam. You must remain in this room until you have
finished the exam.
All your solutions must be written clearly and understandably. Always give an explanation
for you final answer (unless explicitly stated otherwise). If you are unable to do a subproblem of
a specific problem, you may still use the result later on for another subproblem. Use the backs of
the pages if necessary. You might find some of the problems quite easy; try to solve these first.
Good luck!
The following are the rules governing formal examinations:
1. Each candidate must be prepared to produce, upon request, a UBCcard for identification.
2. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambi-
guities in examination questions.
3. No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the
scheduled starting time, or to leave during the first half hour of the examination.
4. Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed
from the examination and shall be liable to disciplinary action:
•having at the place of writing any books, papers or memoranda, calculators, computers, sound or image
players/recorders/transmitters (including telephones), or other memory aid devices, other than those
authorized by the examiners;
•speaking or communicating with other candidates; and
•purposely exposing written papers to the view of other candidates or imaging devices. The plea of
accident or forgetfulness shall not be received.
5. Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and
must not take any examination material from the examination room without permission of the invigilator.
6. Candidates must follow any additional examination rules or directions communicated by the instructor or
invigilator.
Problem Out of Score
1 16
2 22
3 18
Problem Out of Score
4 14
5 14
6 16
Total 100
pf3
pf4
pf5
pf8
pf9
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Partial preview of the text

Download Proof Suffices - Mathematics - Exam and more Exams Mathematics in PDF only on Docsity!

Math 312, Section 101 Final Exam December 5, 2008 Duration: 150 minutes

Please identify yourself in ink with name, student number and signature.

Name: Student Number: Signature:

Do not open this test until instructed to do so! This exam should have 10 pages, including this cover sheet. No textbooks, notes, calculators, or other aids are allowed. Turn off any cell phones, pagers, etc. that could make noise during the exam. You must remain in this room until you have finished the exam.

All your solutions must be written clearly and understandably. Always give an explanation for you final answer (unless explicitly stated otherwise). If you are unable to do a subproblem of a specific problem, you may still use the result later on for another subproblem. Use the backs of the pages if necessary. You might find some of the problems quite easy; try to solve these first. Good luck!

The following are the rules governing formal examinations:

  1. Each candidate must be prepared to produce, upon request, a UBCcard for identification.
  2. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambi- guities in examination questions.
  3. No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination.
  4. Candidates suspected of any of the following, or similar, dishonest practices shall be immediately dismissed from the examination and shall be liable to disciplinary action: - having at the place of writing any books, papers or memoranda, calculators, computers, sound or image players/recorders/transmitters (including telephones), or other memory aid devices, other than those authorized by the examiners; - speaking or communicating with other candidates; and - purposely exposing written papers to the view of other candidates or imaging devices. The plea of accident or forgetfulness shall not be received.
  5. Candidates must not destroy or mutilate any examination material; must hand in all examination papers; and must not take any examination material from the examination room without permission of the invigilator.
  6. Candidates must follow any additional examination rules or directions communicated by the instructor or invigilator.

Problem Out of Score 1 16

2 22

3 18

Problem Out of Score 4 14

5 14

6 16

Total 100

1.[16 pts] For each of the following statements, indicate if it holds for every a, b, c ∈ Z> 0 (if so, a simple ‘true’ without a proof suffices, if not, a ‘false’ together with a counterexample is expected).

(i) If a|b + c, then a|b and a|c. (ii) If a is even, then a is not a prime. (iii) If a is odd, then φ(2a) = φ(a). (iv) If 4 a ≡ 6 (mod 10), then a ≡ 4 (mod 10). (v) If gcd(a, b) = 1, then ab−^1 ≡ 1 (mod b). (vi) If gcd(a, b, c) = 1, then φ(abc) = φ(a)φ(b)φ(c).

(a) [6 pts] Determine, using no moduli other than 111 in your final answer, all integers x that satisfy the following linear congruence. 21 x ≡ 6 (mod 111). (b) [6 pts] Find the least nonnegative integer x that satisfies the following system of linear congruences. x ≡ 998 (mod 999) x ≡ 999 (mod 1000) x ≡ 1000 (mod 1001). (c) [6 pts] Determine all integers x that satisfy the following system of linear congruences. x ≡ 3 (mod 6) x ≡ 1 (mod 10).

(a) [4 pts] Let b ∈ Z> 0. Explain what a pseudoprime to the base b is. (b) [6 pts] Prove that 1729 = 7 · 13 · 19 is a Carmichael number. (Hint: 1728 = 6 · 288 = 12 · 144 = 18 · 96 .) (c) [4 pts] Show, without using the explicit prime factorization of 1729 , but using the follow- ing congruences instead, that 1729 is composite. 218 ≡ 1065 (mod 1729) 236 ≡ 1 (mod 1729).

  1. The purpose of this problem is to prove the following theorem.

Theorem 1. For all a, m ∈ Z> 0 we have am^ ≡ am−φ(m)^ (mod m).

Let a, m ∈ Z> 0. For m = 1, the theorem holds trivially, so we assume from now on that m > 1 and write its prime-power factorization as m = pe 11... pe kk for different primes p 1 ,... , pk, exponents e 1 ,... , ek ∈ Z> 0 and some k ∈ Z> 0. Let i ∈ { 1 ,... , k} and focus on the prime power pe i iin the prime-power factorization of m.

(a) [4 pts] Prove that if pi - a, then pe i i|aφ(m)^ − 1. (Hint: Use Euler’s theorem and prove/use that φ(pe i i)|φ(m).) (b) [4 pts] Proof that if pi|a, then pe i i|am−φ(m). You might want to do this as follows.

  • Prove that pe i i−^1 |m − φ(m).
  • Prove that m − φ(m) ≥ pe i i−^1.
  • Prove that pe i i|pm i −φ(m)(here you may use without proof that pe i i−^1 ≥ ei). (c) [4 pts] Prove that pe i i|am^ − am−φ(m). (Hint: combine the results of (a) and (b).) (d) [4 pts] Complete the proof of Theorem 1 above.

3 extra pages to write solutions on