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# hw05 - 1 Problem 5 Find the smallest positive integer with...

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In the Name of God Fall 2005 Discrete Mathematics Homework 5 Due: Aban 22 Problem 1. For every positive integer p , we consider the equation 1 x + 1 y = 1 p . We are looking for its solutions ( x,y ) in positive integers, with ( x,y ) and ( y,x ) being considered different. Show that if p is prime, then there are exactly three solutions. Otherwise, there are more than three solutions. Problem 2. Find the smallest integral solution of the following system of linear congruences x 5 ( mod 12) x 17 ( mod 20) x 23 ( mod 42) Problem 3. Solve the following system of congruences. x 2 2 ( mod 7) x 2 3 ( mod 11) x 2 4 ( mod 13) Problem 4. Show that in any set of n + 1 positive integers not exceeding 2 n there must be two that are relatively prime.

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Unformatted text preview: 1 Problem 5. Find the smallest positive integer with the property that, if you move the ±rst digit to the end, the new number is 1.5 times larger than the old one. Problem 6. Two players A and B alternatively take chips from two piles with a and b chips, respectively. Initially a > b . A move consists of taking from a pile a multiple of the other pile. The winner is the one who takes the last chip in one of the piles. Show that if a > 2 b , then the ±rst player A can force a win. Problem 7. Prove that n ( n + 1) divides 2(1 k + 2 k + · · · + n k ) for odd k . 2...
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