Problems for Number Theory September 16, 1998

30. Count the number of divisors for every integer from 1 to 400.

31. Find a simple formula for the number of divisors of n.

32. A number is called highly composite if it has more divisors than any smaller number. Find all highly composite numbers less than 400.

33. Characterize the highly composite numbers.

34. A number is deficient if it is greater than the sum of its divisors. A number is abundant if it is less than the sum of its divisors. Find some abundant and deficient numbers.

35. Are all odd numbers deficient?

36. Formulate a condition that guarantees abundance.

37. Formulate a condition that guarantees deficiency.

38. A Pythagorean triple is a set of positive integers, a, b, c, satisfying a^{2} + b^{2} = c^{2}. That is, they are integers which could be the lengths of the sides of a right-angled triangle. A Pythagorean triple is called primitive if gcd(a, b, c) = 1. Find some examples of primitive Pythagorean triples (hereinafter, PPT).

39. If (a, b, c) is a PPT, then a and b have different parity. Moreover, each pair from (a, b, c) are relatively prime.

40. Characterize the entire collection of PPTs.

41. The radius of the inscribed circle of a Pythagorean triangle is always an integer. (A Pythagorean triangle is right-angled and all sides have integer lengths.)

42. Pythagorean triangles have integer areas. Find a pair of such triangles that have the same area.

43. Find all such pairs.

44. Prove that congruence modulo n is an equivalence relation.

45. If a ≡ b (mod n) and c ≡ d (mod n), then a + c ≡ b + d (mod n) and ac ≡ bd (mod n).

46. If a ≡ b (mod n), then a^{k} ≡ b^{k} (mod n) for any positive integer k.

47. A sort of converse: If a^{k} ≡ b^{k} (mod n) for some k > 0, then a ≡ b (mod n).

48. The actual converse: If a^{k} ≡ b^{k} (mod n) for all k > 0, then a ≡ b (mod n).

49. Evaluate 1! + 2! + 3! + 4! + … + 100! (mod 12).

50. Devise and prove a cancellation law for congruence modulo n. That is, is it true that ca ≡ cb (mod n) implies that a ≡ b (mod n)?

51. Evaluate 41^{65} modulo 7.

52. If a is odd, then a^{2} ≡ 1 (mod 8).

53. For any a, a^{3} ≡ a (mod 6).

54. Evaluate 2^{44} (mod 89).

55. Find all solutions to 7x ≡ 5 (mod 17).

56. Find all solutions to 9x ≡ 5 (mod 21).

57. Find all solutions to 9x ≡ 15 (mod 21).

58. State and prove the best theorem(s) you can generalizing the last three results.

59. Find a number which leaves the remainders 2, 3, 2, when divided by 3, 5, 7, respectively.

60. Find a number which leaves the remainders 1, 2, 3 when divided by 3, 5, 6 respectively.

61. When is it possible to solve the system of congruences x ≡ a_{i} (mod n_{i})?