Definition of Cute :O

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 that$$3^2=9,~~ 33^2 = 1089,~~ ,33^2=110889, ~~ 333^2=11108889 $$and the pattern continues (can be proved by induction )then notice that $s(3^2)=9,s(33^2)=18 , s(33^2)=27,s(33^2=)=36$, and so on . So any number of the form $0 \mod 9$ belongs to this set .

part 2- Notice that$$1^2=1,~~19^2 = 361,~~ 199^2=39601,~~ 1999^2=3996001$$ and the pattern continues (can be proved by induction )then notice that $s(1^2)=1, s(19^2)=10 ~~, s(199^2)=19 ~~,s(1999^2=)=28 $ and so on . also $ ..$ So any number of the form $1 \mod 9$ belongs to this set .

part3 - Notice that$$2^2=4,~~ 29^2 = 841, ~~299^2=89401,~~ 2999^2=8994001$$ and the pattern continues (can be proved by induction )then notice that $s(2^2)=4 ,s(29^2)=13 , s(299^2)=22 , s(2999^2=)=31$ and so on. So any number of the form $4 \mod 9$ belongs to this set.

part 4- Notice that$$49^2 = 2401,~~499^2=249001,~~ 4999^2=24990001$$ and the pattern continues (can be proved by induction )then notice that $s(49^2)=7 , s(499^2)= 16,s(4999^2=)=25$ and so on . So any number of the form $7 \mod 9$ belongs to this set.

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!

Oh and pause pause.. If you have time then do subscribe to do this blog. Click the three rows thingy in the Right top which is white in colour, then follow, I have got 8 followers till now. So yayyy!!

Sunaina💜

Comments

Sunanda naikApril 2, 2021 at 1:00 PM
Nice! Good Job . Now study for boards 👍
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