2016...2016

True or false?

There exist a positive integer n n , where 2017 201620162016 2016 Number of 2016 = n 2017\mid \underbrace{201620162016\dots 2016}_{\text{Number of}\space 2016=n}

True False

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1 solution

Áron Bán-Szabó
Jul 31, 2017

Consider the number-table below

2016 20162016 201620162016 20162016 2016 Number of 2016 = 2018 2016 \\ 20162016 \\ 201620162016 \\ \dots \\ \underbrace{20162016\dots 2016}_{\text{Number of}\space2016=2018}

Between these 2018 2018 numbers, there will be two, which make the same remainder when it is divided by 2017 2017 (because the possible values of the remainder are 0 , 1 , 2 , 3 , 4 , , 2016 0,1,2,3,4,\dots,2016 ). The differenc between these numbers is: 20162016 2016 Number of 2016 = k 20162016 2016 Number of 2016 = m = 20162016 201600 00 Number of 2016 = k m = 2016 2016 1 0 l \underbrace{20162016\dots 2016}_{\text{Number of}\space 2016=k}-\underbrace{20162016\dots 2016}_{\text{Number of}\space 2016=m}=\underbrace{20162016\dots 201600\dots 00}_{\text{Number of }\space 2016=k-m}=2016\dots 2016*10^{l} Since gcd ( 1 0 l , 2017 ) = 1 \text{gcd}(10^l,2017)=1 , 2017 20162016 2016 Number of 2016 = k m 2017\mid \underbrace{20162016\dots 2016}_{\text{Number of}\space 2016=k-m}

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