Putnam TNG - Number Theory

1: Do there exist 1,000,000 consecutive integers each of which contains a repeated prime factor?

2: Show that for each positive integer n,

n! = _i=1∏ⁿ lcm{1, 2, . . . , ⌊n/i⌋}.

3: Find all integer solutions to 15x² - 7y² = 9.

4: Suppose I have a set of positive integers such that each is less than 2003, and the least common multiple of any pair of them is greater than 2003 show that the sum of their reciprocals is less than 2.

5: Suppose a and b are positive integers. Find all integer values of

(a² + ab + b²/ab - 1)

6: Suppose n is a positive integer. Determine all values of n such that n⁵ +n⁴ +1 is prime.

7: Do there exist positive integers a and b with b > a + 1 such that for every integer k with a < k < b, either gcd(a, k) > 1 or gcd(b, k) > 1?

8: Let d(n) be the largest odd number which divides a given number n. Suppose that D(n) and T(n) are defined by

D(n) = d(1) + d(2) + · · · + d(n)

      T(n) = 1 + 2 + · · · n.

prove there exist infinitely many positive numbers such that 3D(n) = 2T(n).