MathDB
Dividing properties

Source: Austrian-Polish 2006 Question 1

August 21, 2006
number theory proposednumber theory

Problem Statement

Let M(n)={n,n+1,n+2,n+3,n+4,n+5}M(n)=\{n,n+1,n+2,n+3,n+4,n+5\} be a set of 6 consecutive integers. Let's take all values of the form ab+cd+ef\frac{a}{b}+\frac{c}{d}+\frac{e}{f} with the set {a,b,c,d,e,f}=M(n)\{a,b,c,d,e,f\}=M(n). Let xu+yv+zw=xvw+yuw+zuvuvw\frac{x}{u}+\frac{y}{v}+\frac{z}{w}=\frac{xvw+yuw+zuv}{uvw} be the greatest of all these values. a) show: for all odd nn hold: gcd⁡(xvw+yuw+zuv,uvw)=1\gcd (xvw+yuw+zuv, uvw)=1 iff gcd⁡(x,u)=gcd⁡(y,v)=gcd⁡(z,w)=1\gcd (x,u)=\gcd (y,v)=\gcd (z,w)=1. b) for which positive integers nn hold gcd⁡(xvw+yuw+zuv,uvw)=1\gcd (xvw+yuw+zuv, uvw)=1?
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