Hey everyone, as I was going through my previous HSO Posts. I saw this beautiful problem which was given to me by Rege sir ( He teaches in IMOTC and INMOTC of Assam). I posted the solution in HSO on April 8 2020. And today I latexed the solution ( After like a year). The solution I got is pretty cute! Try it once!
Problem:- Determine the set A={s(n^2)| n is a positive integer}. (1 = s(1) = s(100), 4 = s(4) = s(121) belong to this set.)
Solution:- Let A={s(n^2)| n is a positive integer}notice n^2 is always 0,1,4,7 \mod 9 .by using the fact that 9| s(a) - a , a is an integer we get that 9| s(n^2) - n^2 which implies s(n^2) is always 0,1,4,7 \mod 9 .
claim - Any number of the form 0, 1, 4, 7 \mod 9 belongs to set A
proof-
part 1- Notice that3^2=9,~~ 33^2 = 1089,~~ ,33^2=110889, ~~ 333^2=11108889
part 2- Notice that1^2=1,~~19^2 = 361,~~ 199^2=39601,~~ 1999^2=3996001
part3 - Notice that2^2=4,~~ 29^2 = 841, ~~299^2=89401,~~ 2999^2=8994001
part 4- Notice that49^2 = 2401,~~499^2=249001,~~ 4999^2=24990001
And we are done!
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Anyways, now I will go and do some NCERT bio ( complete full syllabus of science today) and I can send my notes too here ( It's very aesthetic though ). Wish me luck!
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Sunaina💜
Nice! Good Job . Now study for boards 👍
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