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1.  How many positive integer divisors of 2004^2004 are divisible by exactly 2004 positive integers?

2.  Assume that a, b, c, and d are positive integers such that a^5 = b^4, c^3 = d^2, and c - a = 19. Determine d - b.

3.  Find the sum of all positive integers a=2^{n}3^{m}, where n and m are non-negative integers, for which a^6 is not a divisor of 6^a.

4.  There is a set of 1000 switches, which are ordered in a row so that each switch is given a distinct rank from 1 to 1000. For example, the i-th switch refers to the switch given rank i. Each switch has four positions, called A, B, C, and D. When the position of any switch changes, it is only from A to B, from B to C, from C to D, or from D to A. Initially each switch is in position A. The switches are labeled arbitrarily with the 1000 different integers (2^x)(3^y)(5^z), where x, y, and z take on the values 0, 1, .... , 9. At step i of a 1000-step process, the i-th switch is advanced one step, and so are all the other switches whose labels divide the label on the i-th switch. After step 1000 has been completed, how many switches will be in position A?

Jul 1, 2020