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Prime Factorization - Number Theory - Exam, Exams of Number Theory

Aligarh Muslim UniversityNumber Theory

This is the Exam of Number Theory which includes Residues Modulo, Nonnegative Integers, Positive Integers, Elements, Prime Divisor, Exist Infinitely, Integer Solution, Congruence etc. Key important points are: Prime Factorization, Remainder, Induction, Non Trivial Factor, Right Angled Triangles, Integer Sides, Given Length, Multiple, Divided, Chinese Remainder Theorem

Typology: Exams

2012/2013

Uploaded on 02/18/2013

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pg score/32
Name:
MATH 506 Number Theory – Final Exam
Monday May 7, 2012
Check that you have all four pages. Show all your work. Assume multiplicativity where appropriate.
1. (4 points) If a= 2413219, b= 235213 then the prime factorization of (a2, b3) =
2. (6 points) (a) Evaluate: φ(3000) =
(b) The remainder when 112402 is divided by 3000 is .
3. (6 points) Use induction to prove that 2n|(2n)!
4. (4 points) A non-trivial factor of 252 + 1 is .
5. (12 points) Find all right-angled triangles with coprime integer sides and base of given length:
(i) 28
(ii) 15.
1
pf3
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pg score/

Name:

MATH 506 Number Theory – Final Exam

Monday May 7, 2012 Check that you have all four pages. Show all your work. Assume multiplicativity where appropriate.

  1. (4 points) If a = 2^4132 19, b = 2^352 13 then the prime factorization of (a^2 , b^3 ) =
  2. (6 points) (a) Evaluate: φ(3000) = (b) The remainder when 11^2402 is divided by 3000 is.
  3. (6 points) Use induction to prove that 2n^ | (2n)!
  4. (4 points) A non-trivial factor of 2^52 + 1 is.
  5. (12 points) Find all right-angled triangles with coprime integer sides and base of given length: (i) 28

(ii) 15.

pg score/

  1. (10 points) (a) If F (n) = ∑ d|n

τ (d)^2 then F (175) =

(b) If σ(n)^2 = ∑ d|n

g(d) then g(5^2 ) =

  1. (12 points) n = 126126000 x 7575752 (a) If n is a multiple of 9 then x =.

(b) If n leaves remainder 2 when divided by 11 then x =.

(c) If 2e||n then e =.

  1. (10 points) (a) The order of 2 mod 17 is.

(b) If the order of b mod m is 30 then the order of b^24 mod m is.

  1. (8 points) Use the Chinese Remainder Theorem to solve the simultaneous congruences x ≡ 3 (mod 5) x ≡ 4 (mod 6) x ≡ 5 (mod 7)

pg score/

  1. (12 points) (a) Use the Euclidean algorithm to compute the greatest common divisor (684,589)

(b) Solve the linear equation 589x − 684 y = 247 or explain why there are no solutions.

(c) Solve the linear congruence 589x ≡ 247 (mod 684) or say why no solutions exist.

  1. (5 points) You have decided to do RSA cryptography with modulus n = 139·179 and encode exponent
  2. Give (but don’t solve) a congruence that you would use to find a decode exponent d.
  3. (12 points) (a) Calculate the continued fraction expansion of 942/ 769

(b) Calculate the continued fraction convergents

(c) A mechanic wants a gear ratio of approximately 942/769. How many teeth should the gear wheels have, assuming that no wheel has more than fifty?