Question

1. (a) Choose 150 integers from this list {1, 2, ..., 298}, prove that there are two integers ni, n2 such that ni|n2 or n2|n1. (b) Let n1, 12, ... , 1201 be integers. Prove there exist three in- tegers ni, nj, nk E {n1, N2, ... , n201} such that 100 can divide the differences between any two of them.

Answer 1

O Sol- we have to choose 10 integers from the list {be, ---2984 that are we have to prove two integers n, and n2 such that hilar nalni we choose nial. then nilke where ne any integer and it than na ln, where ni any integer done. is 12=1 so we are K, is niln2 = n2 = kini, NOW, Integer any a) heln, =sn = K2 M2, K2 is any Integer - two integey scalar o.) ta Thus if we close multiple of we are done. (nin2 chosen from 150 no prime number we get " as dirish. wether so ut nal and ne=P (prime number) from 150 db {1, 2 - 2984 then nil12. integen If nfl Then n, can be scalar multiple of na, nizke, k integer that is

as no (8) take nalni for composite number, we get 2813 scalar multiples. Which implies hilne nalno Hence proved be integers Given nina, 201 If we choose n. 21, n.oo 100, 101 101 and 1201²201 Then 120,-1,200 bio-n, = 100 1201-Noi - 100 TOO which au divided by 1;,n; MKE {hin, -n2013 let NOW, then oo /no-n; = killOO %), » ninj ning + 4,100 100/n; nk n; na K2 100 ) nj = nkt K2 x 100

100 10; na =)n, -nk = K₂ 100 => ni Enkt Kz 100 arl interger K1, K2, K3 if we choose ni,n; OK E {ny, ---N2014 in such a way that ni =n; +4100 ni = Okt (kit 200 nj= net k2 100 and nk such away that we choose Klik4 kg are in no, ni, nk are in {n--2014: blw them. Then 100 can diride the difference so y nunsink e {nicze -- Day! such that too can them. divide the difference between As nin ni define above exists in {n, in-hoy can devide - the difference atleast one, so 100 between them proved

In set of 201 number which are integey we atleast which are either get three numbey multiple of 100 ) their difference is a multiple do sum (in case their of 100 Cors negative integer) taking two at a time یا 100 is multiple so, in {nina 20ly we get atleast three numbers ning, nic such that nin;= look, nj-nk=look2 ne-n; = 100k 3 in such a way Kik21K3 are integey taking that ni, n;, nce{--thony Hence proved Thank you