# hw5 - Let A denote the sum of the decimal digits of 4444...

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Putnam Exam Seminar Assignment 5 Fall 2010 Due October 11 Exercise 1. Determine whether or not the matrix 117 218 344 511 1007 101 800 911 578 113 1212 14 4216 178 2013 516 19 2114 104 3416 789 534 114 472 300 has an inverse. Exercise 2. Determine the number of pairs of positive integers ( m,n ) that satisfy the equation 19 m + 102 + 8 n = 2010. Exercise 3. Consider the set { 2 , 5 , 13 } . Show that if D 6∈ { 2 , 5 , 13 } then there exist A,B ∈ { 2 , 5 , 13 ,D } so that AB - 1 is not a perfect square. Exercise 4.
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Unformatted text preview: Let A denote the sum of the decimal digits of 4444 4444 and let B be the sum of the decimal digits of A . Find the sum of the decimal digits of B . Exercise 5. Prove that every positive integer has a multiple whose decimal representation includes all ten digits. Exercise 6. Suppose p is an odd prime. Prove that p X j =0 ± p j ²± p + j j ² ≡ 2 p + 1 (mod p 2 ) . [Putnam Exam 1991, B-4]...
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## This note was uploaded on 02/29/2012 for the course MATH 1190 taught by Professor Ryandaileda during the Fall '10 term at Trinity University.

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