Elements of Number Theory — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

1. (a) If n is odd integer, show that nt + 4n? +11 is of the form 16k, for some integer k. (b) By using division algorithm show that the cube of any integer has one of the forms 9k, 9k + 1, 9k + 8, for some integer k. (c) Prove that god (a,b) = 1, then god (a+b, ab) = 1. 2. (a) State Fundamental Theorem of Arithmetic and prove that for any integer k # 0, ged (ka, kb) = |k| (b) Show that 41 divides 220 - 1. (c) Solve the following simultaneous congruences : x = 2(mod3), x = 3(mod5), x = 2(mod7). 3. (a) If ged(a, 35) = 1, show that al? = 1(mod35). (b) State Wilson's Theorem and use it to find the remainder when 15! is divided by 17. (c) Prove that (n) is an odd integer if and only if n is a perfect square. 4. (a) Find the highest power of 5 dividing 1000! and highest power of 2 dividing 50!. (b) Show that (n) is an even integer for n>2 and calculate (5040): (c) Prove that: 10-1 + 2P-1 + 3P-1+... + (p - 1)P-1 = -1 (mod p), where p is an odd crime (a) Employ Hills Cipher to encrypt the message BUYNOW using the congruences (b) Prove that the integer n = 21°(2"-1) is not a perfect number by showing that o(n) # 2n. (c) Show that the Mersenne number M23 is not a prime number. 6. (a) Prove that the Eermat number Fs = 22+1 is divisible by 641. (b) If the Caesar cipher produced KDSSBELUWKGDB, what is the plaintext message? (c) Show that the sum of the squares of the first n Fibonacci numbers is given by the formula