Elements of Number Theory — DU B.Sc. (Prog) Physical Sciences courses pertaining to Mathematical Sciences Previous Year Question Papers

ELEMENTS OF NUMBER B.Sc. (Physical Sc./Math Sc.) I. (a) Use the Euclidean Algorithm to obtain integers x and y satisfying gcd (x, y) = 24x + 138y (b) A farmer purchased 100 head of livestock for a total cost of $4000. Prices were as follows: calves, $120 each; lambs, $50 each; piglets, $25 each. If the farmer obtained at least one animal of each type, how many of each did he buy? (c) State Fundamental Theorem of Arithmetic. Express 4 725 & 17 460 in their canonical forms and hence find gcd(4725, 17460) 2. (a) (i) Let m denote all positive divisors of 12, then verify that u I U12 : where un denotes the nth Fibonacci number. (ii) Find the remainder when 250 is divided by 7. (b) Solve the set of simultaneous congruences: x = 2 (mod 3) , x = 3 (mod 5) , x = 2 (mod 7) (c) Find the solutions of the system -of congruences: 3x + 4y = 5 (mod 13) and 2x + Sy = 7 (mod 13) 3. (a) State & Prove Fermat's theorem. Verify 17 divides (b) State Wilson's theorem. Comment on its converse. Find the remainder when 2(26) ! is divided by 29. (c) Find the highest power of 7 dividing 2000! 4. (a) Define r and cr functions. Find cr ( 180) and (b) Prove that r (n) is an odd integer if and only if n is a perfect square. (c) (i) Find the unit digit of 3100 by the use of Fermat's theorem. (ii) For any integer a, verify that a5 and a have the same unit digit. 5. (a) Define Fermat Number and Mersenne number. Verify that M 11 is a composite number (ii) Prove that 3 <p (3n) - 3 <p (n) if and only if ( c) If Caesar cipher produced "KDYHD QLFHD DB "text message than what will be plaintext message. 6. (a) Use the Hill Cipher to encrypt the message "SLOW AND STEADY". (b) Encrypt the message "ONE IF BY DAWN" using the letter "K" as the seed. (c) Prove that the integer n = 2 10 (2 11 - 1) is not a perfect number by showing that a(n) -:,; 2n-