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ClassRoom 42 Anant Vyawahare At Right Angles |

Vol. 5,No.2,July 2016 ec u hieo notation, of choice our hence tefantrlnme.Frexample: For number. natural a itself • • The Introduction Root Digital The ubr n oo,tl igedgtnme sotie.I sdenoted is It obtained. is resulting number the digit of single digits by a the till of on, sum so the and computing number, then digits, its of sum h olwn rtmtcpoete a eesl eiid Here easily verified. be can n properties following root The digital the of properties arithmetical Ten basis the examine will presently. We procedure results. this was their for idea check the to devices, beenknownfor accountants computer has by of used development number the natural Before ofa time. root some ofdigital concept The Property Property , square, , sixthpower, modulo Keywords: Digitalroot, remainder, Beejank, sumofdigits,, , B B m B ( ( , iia root digital ( 4359 79 k n , ) )= p nVdcmteais h iia oti nw as known is root digital the mathematics, Vedic In . ,... 1. 2. )= B ( scerytu,frif for claim true, the clearly But is claim. the prove will recursively, forward h iisof digits the n If opoeti,i ufcst hwta o any positive for that to show suffices it this, prove To h ifrnebtennandB between difference The eoepstv nees(nesohriespecified). otherwise (unless positive denote 7 n B − fantrlnumber natural a of 1 ( + = 4 ≤ B + 9 a ( )= 0 n n h ifrnebetween difference the n, 3 )= + ≤ + 10a B hnB then 9, n 5 ( o oennngtv nee k. integer non-negative some for 9k 16 samlil f9 hsse,carried step, This 9. of multiple a is + 1 B )= + 9 ( )= n 10 ) ( 1 oeta h iia otof root digital the that Note . n 2 n a + B )= 2 ( sotie ycmuigthe computing by obtained is 21 6 + = n. )= 10 ( ;ta is, that 7; n 3 a ) 3. 3 n samlil of multiple a is + n h sum the and , ··· 1

B ( 79 )=

Beejank s i.e., 9; ( n 7. n ) of is ;

Vol. 5,No.2,July 2016 |

At Right Angles 42

ClassRoom 42 43 Anant Vyawahare At Right Angles At Right Angles | |

Vol. 5,No.2,July 2016 Vol. 5,No.2,July 2016 ec u hieo notation, of choice our hence tefantrlnme.Frexample: For number. natural a itself ubr n oo,tl igedgtnme sotie.I sdenoted is It obtained. is resulting number the digit of single digits by a the till of on, sum so the and computing number, then digits, its of sum The Introduction Root Digital The • • h olwn rtmtcpoete a eesl eiid Here easily verified. be can n properties arithmetic following root The digital the of properties arithmetical Ten basis the examine will presently. We procedure results. this was their for idea check the to devices, beenknownfor accountants computer has by of used development number the natural Before ofa time. root some ofdigital concept The Property Property triangular number, square, cube, sixthpower, modulo Keywords: Digitalroot, remainder, Beejank, sumofdigits,natural number, , B B m B ( ( , iia root digital ( 4359 79 k n , ) )= p nVdcmteais h iia oti nw as known is root digital the mathematics, Vedic In . ,... 1. 2. )= B ( scerytu,frif for claim true, the clearly But is claim. the prove will recursively, forward of digits the n If opoeti,i ufcst hwta o any positive for that to show integer suffices it this, prove To h ifrnebtennandB between difference The eoepstv nees(nesohriespecified). otherwise (unless integers positive denote 7 n B − fantrlnumber natural a of 1 ( + = 4 ≤ B + 9 a ( )= 0 n n h ifrnebetween difference the n, 3 )= + ≤ + 10a B hnB then 9, n 5 ( o oennngtv nee k. integer non-negative some for 9k 16 samlil f9 hsse,carried step, This 9. of multiple a is + 1 B )= + 9 ( )= n 10 ) ( 1 oeta h iia otof root digital the that Note . n 2 n a + B )= 2 ( sotie ycmuigthe computing by obtained is 21 6 + = n. )= 10 ( ;ta is, that 7; n 3 a ) 3. 3 n samlil of multiple a is + n h sum the and , ··· 1 B ( 79 )= Beejank s i.e., 9; ( n 7. n ) of is ;

Property Property Property Property 3. 5. 4. 6. ealta a that Recall iio of itself. 3 is 3 Remark of multiple a is which number prime only n then and let example, For 10 coefficients the Since aclto ycsigotnns”Tu,w a ucl hwta thecomputation a that “checking of show method quickly traditional can the we of Thus, basis nines.” the out forms casting 3. which by Property 3 calculation from Property again is It once follows it Comment. you; to this of proof the leave We follow fromthe They will 3. integers If Property using assertion: ingeneral general more befollowing proved may relations two The eane ne iiinb ,and 9, by division under remainder fni utpeof multiple a is n If opoeteaoecam ecnie h ifrn om htatinua number atriangular that forms different the consider n we claim, above the prove To Corollary utb norc,bcuetedgtlro fteepeso ntelf ieis side left the on expression the of root digital the because incorrect, be must eane ntedvso n division the in remainder ,then 9, o,since For, exceeding number prime A hl h iia oto h xrsino h ih iei 4. is side right the on expression the of root digital the while m o l ar m pairs all For 2. and Property 1 Property from follows This h iia oto raglrnme soeo h followingnumbers: one of the is number ofatriangular root digital The to find a counterexample for this claim. for acounterexample find to − = + s m n s ( ( n ( n = n ) m )= h olwn ttmn ok luil butis plausible looks statement following The : − samlil f9. of multiple a is B 29, n n + : ( h ifrnebtennandB between difference The s and − ( B B a 1 raglrnumber triangular n mn 0 , ( ) ( B )=( m / fitgr,tefloigrltosaetrue: are relations following the integers, of n + B ( B a ae eedn ntermidrthat remainder the on depending have, can 2 m )+ ( n = ( m a ) n hnB then 9, = 1 ) ) samlil f3 if 3, of multiple a is 10 0,so 204, + B B seult h raetcmo iio of divisor common greatest the to equal is 12, ( ( B 34567 a − n n ( − 2 )= )) )= )) m B 3 n B 1 + ÷ ,10 1, ( ( ) ( anthv iia oteulto equal root digital a have cannot = + mn B m a n B a 9. 1 ( )= 3 ( × B n 7 Then 17. B m + mn + + )= )= ( ( 2 a h form the has 3 23456 3 + n ( − ··· fni o utpeof multiple a not is n If 9. )= )= + 10 × n n B B ,10 1, , )= n therefore: and , 8 2 8 n ( ( )= )= − ′ B B B B B = B ieielaetesm eane ne iiinby division under remainder same the leave likewise ( ( ( ( ( ( 3 2, 1 m m ( n m m n m 810802552 ) − B ) B ) )+ ) m ′ ′ B a )= ( ( samlil of multiple a is B n + samlil f3 hns utbe must so then 3, of multiple a is ( 11 2 ′ 1 m 24 ′ ( mn ) and ,... + n ( B n )= . )= m )) ′ B ( ) )= ( n ( 10 , + . n )) 12 r l utpe f9 tflosthat follows it 9, of multiples all are ′ 2 6 3 1 r uhthat such are , )= .Nwosrethat: observe Now 6. = = ) − / not Vol. 5,No.2,July 2016 Vol. 5,No.2,July 2016 where 2 B B 1 and 3 ( ) ( hnB then 9, 3. m mn re Tegets common greatest “The true: 3, a m 3 + 6 ) B + evsudrdvso y6. by division under leaves . or B ( n m n . ··· ) ( m ) , n 9. 1 ( sapstv integer. positive a is )= n . eivt you invite We 9.” and , , n m 3 ) , ′ seult the to equal is 6 ev h same the leave B , ( 9. 17 | | B

At Right Angles At Right Angles ( )= 7 n the and n; × .Also, 8. 2 )= 5, 42 43 The different forms for m are: 6k,6k + 1, 6k + 2, 6k + 3, 6k + 4, 6k + 5. If m = 6k or 6k + 2 or 6k + 3 or 6k + 5, then the product m(m + 1)/2 is a multiple of 3; hence the of m(m + 1)/2 will be one of the numbers 3, 6, 9. If m = 6k + 1 or 6k + 4, then m(m + 1)/2 is of the form 9m + 1; hence its digital root is 1. Expressed negatively, the above result yields a useful corollary. Corollary. A triangular number which is not a multiple of 3 has digital root equal to 1. Property 7. If n is a perfect square, then B(n) 1, 4, 7, 9 . ∈{ } We consider the different forms that a perfect square n = m2 can have, depending on the remainder that m leaves under division by 9. The different forms for m are: 9k,9k 1, ± 9k 2, 9k 3, 9k 4. By squaring the expressions and discarding multiples of 9, we find ± ± ± that B(n) 1, 4, 7, 9 in each case. ∈{ } Property 8. If n is a perfect cube, then B(n) 1, 8, 9 . Digital Roots of

∈{ } ClassRoom The same method may be used as in the case of the squares. Property 9. If n is a perfect , then B(n) 1, 9 . Perfect Numbers ∈{ } If n is a perfect sixth power, then it is a perfect square as well as a perfect cube; hence B(n) ∈ 1, 4, 7, 9 as well as B(n) 1, 8, 9 . This yields: B(n) 1, 9 . (A nice application of { } ∈{ } ∈{ } set intersection!) Corollary. A perfect sixth power which is not a multiple of 3 has digital root equal to 1. Property 10. If n is an even other than 6, then B(n)=1. Recall that a perfect number is one for which the sum of its proper equals itself. The first few perfect numbers are: t was claimed in the article by Anant Vyawahare on Digital 6; 28; 496; 8128; 33550336. The digital roots of these numbers are: Roots that a perfect number other than 6 has digital root 1 (Property 10 of that article). We now provide a proof of this ( )= ( )= , B 28 B 10 1 Iclaim. ( )= ( )= , B 496 B 19 1 But, first things first; let us start by defining the basic motions B(8128)=B(19)=1, and related concepts. ( )= ( )= , B 33550336 B 28 1 Perfect numbers. Many children discover for themselves the and so on. following property of the integer 6: the sum of its proper divisors is equal to the number (1 + 2 + 3 = 6). Noticing such a This result is far from obvious and will need to be proved in stages. The full proof isgivenin property, they may naturally wonder about the existence of more the addendum at the end of this article. such integers. Two millennia back, the Greeks decided that such a property indicated a kind of perfection, and called such Concluding remark. The notion of digital root has been known for many centuries. As described inthis numbers perfect. (This is of course the English translation ofthe article, there is a simple number theoretic basis for the notion. The simplicity of the topic makes itan word used by the Greeks; other translations could be: complete, attractive one for closer study by students in middle school and high school. It certainly needs to be better ideal.) So 6 is a perfect number (and it is obviously the smallest known than it is at present. perfect number). Students will naturally ask what we should call numbers which are not perfect, i.e., other than simply calling them imperfect! This line of thinking gives rise to the following definitions. ANANT W. VYAWAHARE received his Ph.D in Number Theory. He retired as Professor, M.Mohota Science College, Nagpur. At present, he is pursuing research on the History of Mathematics. He has a special interest in Vedic Mathematics and in the mathematics in music. Keywords: Perfect number, , , , factor theorem

4544 At Right Angles || Vol. 5, No. 2, July 2016 Vol. 5, No. 2, July 2016 || At Right Angles 4544

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