Advaces i Pure Mathematics, 04, 4, 95-30 Published Olie Jue 04 i SciRes. http://www.scirp.org/joural/apm http://dx.doi.org/0.436/apm.04.46039 O Some Properties of Digital Roots Ilha M. Izmirli Departmet of Statistics, George Maso Uiversity, Fairfax, USA Email: iizmirl@gmu.edu Received 3 March 04; revised 4 April 04; accepted 9 May 04 Copyright 04 by author ad Scietific Research Publishig Ic. This wor is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY. http://creativecommos.org/liceses/by/4.0/ Abstract Digital roots of umbers have several iterestig properties, most of which are well-ow. I this paper, our goal is to prove some lesser ow results cocerig the digital roots of powers of umbers i a arithmetic progressio. We will also state some theorems cocerig the digital roots of Fermat umbers ad star umbers. We will coclude our paper by a iterestig applicatio. Keywords Digital Roots, Additive Persistece, Perfect Numbers, Mersee Primes, Fermat Numbers, Star Numbers. Itroductio The part of mathematics that deals with the properties of specific types of umbers ad their uses i puzzles ad recreatioal mathematics has always fasciated scietists ad mathematicias (O Beire 96 [], Garder 987 []. I this short paper, we will tal about digital roots a well-established ad useful part of recreatioal mathematics which materializes i as diverse applicatios as computer programmig (Trott 004 [3] ad umerology (Ghaam 0 [4]. As will see, digital roots are equivalet to modulo 9 arithmetic (Property.6 ad hece ca be thought of as a special case of modular arithmetic of Gauss (Dudley, 978 [5]. Let us start out by the followig existece theorem: Theorem.. Let be a atural umber ad let s( deote the sum of the digits of. I a fiite umber of steps, the sequece s, s( s, s( s( s, becomes a costat. Proof. Let = 0 d + 0 d + + 0d + d, where for ay 0 j, 0 9. This implies that 0 = + + + + 0 s d d d d If is a oe digit umber, that is if d = d = = d = 0, the, 0 d j s = d = is the required costat. How to cite this paper: Izmirli, I.M. (04 O Some Properties of Digital Roots. Advaces i Pure Mathematics, 4, 95-30. http://dx.doi.org/0.436/apm.04.46039
Else, at least oe of d, d,, d is positive ad, s repeatedly applyig the s operator, we will get a decreasig sequece of umbers. Oce a term of this sequece becomes a sigle digit umber, from the o the sequece will remai a costat. Defiitio.. Let ρ deote the costat value the sequece s, s( s, s( s( s, coverges to. We call ρ the digital root of. Here are some simple properties that follow immediately from this defiitio: Property.. ρ = 9 9, where x stads for the geatest iteger less tha or equal to x. Property.. ρ 9 if mod 9 = 0 = mod 9 otherwise Property.3. ρ( m ρ( ρ( m ρ Property.4. ρ( m = ρ( ρ( m ρ. Property.5. ρ( ρ = ρ. + = +. Property.6. This 9 9 symmetric matrix Table, which is formed by replacig the umbers i a regular 9 9 multiplicatio table by their digital roots, is referred to as a Vedic square. Vedic squares have bee used extesively to create geometric patters ad symmetries, ad eve musical compositios by highlightig specific umbers. For more iformatio see Pritchard (003 [6]. Closely related to the cocept of digital roots is that of additive persistece, which is defied as the umber of (additive steps required to obtai its digital root. We will deote the additive persistece of a oegative iteger by AP (. Clearly, for ay sigle digit umber the additive persistece is. cause we eed two steps to obtai ρ ( 34568 : < Step. 3+ 4+ 5+ 6+ 8= 6 Step. + 6= 8 The smallest umber with additive persistece of, is 999 9 followed by 9 s. For more iformatio o additive persistece see Hide (974 [7]. Some Well-Kow Results Propositio.. Digital root of a square is, 4, 7, or 9. By Property., the digital root of x is ( x = ( ( x ( x ρ ρ ρ ρ which ca oly be ρ( ρ( ρ( ρ( ρ( ρ( ρ( ρ( ρ ( 9 = 9. AP 34568 =, be- =, = 4, 3 = 9, 4 = 7, 5 = 7, 6 = 9, 7 = 4, 8 =, Propositio.. Digital root of a perfect cube is, 8 or 9. Proof is similar to the oe give above. Propositio.3. Digital roots of the powers of a atural umber x form a cyclical sequece. This cycle is the same for all umbers x+ 9, where is ay atural umber: This follows because for ay x, 0 x 9 ad for ay two atural umbers ad r (( 9 r r x+ = ( x = ( ( x ( x ρ ρ ρ ρ ρ We ca use Table to compute digital roots of powers of large umbers. For example, 54 54 ρ ρ 4, 764 = 4 = Propositio.5. Digital root of a eve perfect umber (except 6 is. Proof. Every eve perfect umber m is of the form 96
Table. The multiplicatio table for digital roots is the familiar modulo 9 multiplicatio table with 0 replaced by 9. 3 4 5 6 7 8 9 3 4 5 6 7 8 9 4 6 8 3 5 7 9 3 3 6 9 3 6 9 3 6 9 4 4 8 3 7 6 5 9 5 5 6 7 3 8 4 9 6 6 3 9 6 3 9 6 3 9 7 7 5 3 8 6 4 9 8 8 7 6 5 4 3 9 9 9 9 9 9 9 9 9 9 9 Table. Digital roots of the powers of a atural umber x form a cyclical sequece. x Digital Roots of Successive Powers of x 0 0,0,9,,, 0,,, 4,8,7,5 3,,,,3,9,9,9, 4,3,,, 4, 7 5,4,3,,5,7,8, 4, 6,5, 4,,6,9,9,9, 7,6, 5,, 7, 4 8,7, 6,, 8 9,8,7, where p is a Mersee prime. By puttig ad,9,9,9,9, p m = x p p =, we have ( m = ( x x = ( x ( x ( x ρ ρ ρ ρ ρ Here, the last equality follows from properties,, ad 3 above. By propositio., ρ ( x = 4, 7,, 7,, ( x ( x ρ ρ = 5,8,,8,, Hece the result follows. To geeralize the cocept of digital roots to ay other base b, oe should simply chage the 9 i the formulas to b. For more iformatio o digital roots see Averbach ad Chei (000 [8]. I the followig sectios, we will prove some results o digital roots of powers of umbers i a arithmetic progressio as well as digital roots of Fermat umbers ad star umbers.. Digital Roots of Powers of Numbers i a Arithmetic Progressio We start with the followig 97
Propositio.. Let, m ad be three cosecutive terms i a arithmetic progressio with commo differece d. Let If d is ot a multiple of 3, the ρ ( x = 9. Proof. Let = m d ad = m+ d. The x = + m + 3 x = + m + = m + md = m( m + d Cosequetly, to prove the propositio, we must prove m( m d 3 6 3. + is divisible 3 by for ay atural umber m ad for ay atural umber d that is ot a multiple of three. If m is divisible by 3, the result follows. So, let us cosider the two cases: Case. m= 3r+. I this case Sice d = 3s±, Case. m= 3r+. I this case Agai, ( + = ( 3 + ( 9 + 6 + + m m d r r r d d = 8s ± s+ ( + = ( 3 + ( 9 + 6 + + 8 ± + = ( 3r+ ( 9r + 6r+ 8s ± s+ 3 = 33 ( r+ ( 3r + r+ 6s ± 4s+ m m d r r r s s ( + = ( 3 + ( 9 + + 4 + m m d r r r d d = 8s ± s+ ( + = ( 3 + ( 9 + + 8 ± + 6 = 3( 3r+ ( 3r + 4r+ 6s ± 4s+ m m d r r r s s Remar. The restrictio o d is ecessary. For example, let = 7, m = 0 ad = 3. The, x = + m + = 343 + 000 + 97 = 3540 ad ρ ( x = 3 Usig the fact that a sum is divisible by a positive iteger if all terms are divisible by a positive iteger we get Theorem.. Let q be a multiple of three. Let,,, q be ay q cosecutive terms of a arithmetic progressio whose commo differece d is ot a multiple of three. Let The, ( x 9 ρ =. For example, let ad x = + + + q =, = 7, 3 =, 4 = 7, 5 = ad 6 = 7 The, x = + + + 6 = 8 + 343 + 78 + 493 + 0648 + 9683 = 3733 ρ 3733 = 9. Agai if d = 3, this does ot hold. As a couterexample, =, = 5, 3 = 8, 4 =, 5 = 3 ad 6 = 6 98
ad x = + + + 6 = 8 + 5 + 5 + 33+ 97 + 4096 = 869 ρ 896 = 7. Corollary.. Let q be a multiple of three. Puttig d =, we get that the sum of the cubes of q cosecutive itegers is divisible by 9. Puttig d =, we get that the sum of the cubes of q cosecutive odd itegers (eve itegers is divisible by 9. Although similar results do ot ecessarily hold for sixth powers, we show that they do for ith powers. I fact, we fid out that the restrictio o d is ot eeded for ith powers. Propositio.. Let,,, 9 be ie cosecutive terms i a arithmetic progressio with commo differece d. Let The, 9 ρ x =. Proof. This follows by writig ad otig that x = + + + 9 9 9 9 = 5 4d = 5 3d 3 = 5 d 4 = 5 d 6 = 5 + d 7 = 5 + d 8 = 5 + 3d 9 = 5 + 4d x= + + + = 9 + 40d + + 44, 708d 9 9 9 9 7 8 9 5 5 5 Usig the fact that a sum is divisible by a positive iteger if all terms are divisible by a positive iteger we get Theorem.. Let q be a multiple of ie. Let,,, q be ay q cosecutive terms of a arithmetic progressio whose commo differece d. Let 9 9 9 x = + + + The, ( x 9 q ρ =. Corollary.. Let q be a multiple of ie. Puttig d =, we get that the sum of the ith powers of q cosecutive itegers is divisible by 9. Puttig d =, we get that the sum of the ith powers of q cosecutive odd itegers (eve itegers is divisible by 9. 3. Digital Roots of Fermat Numbers As is well-ow, a Fermat umber F is defied as F = + For computatioal purposes the followig recursio formula is useful: Theorem 3.. For, F = F F + Proof. Sice ad for, F F = + = + 99
F = + F = = F F = F F F + ad the formula follows. Ispectio of the first few Fermat umbers F 0 = 3, F = 5, F = 7, F 3 = 57, F 4 = 655373, F 5 = 49496797 shows that for ρ is 5 if is odd ad 8 if is eve. I fact, this is ideed true for all : Theorem 3.. Let F be the F, th Fermat umber. The, ρ ( F 5 if is odd = 8 if is eve Proof. Proof is by iductio. Clearly, the claim is true for. Assume it is true for. The, Suppose is odd. The Suppose is eve. The 4. Digital Roots of Star Numbers ρ ( F = ρ( F F + ρ( ρ( F ρ( F ρ ρ( ρ( ρ( F ρ( F ρ( F = + = + ρ = 5 5 + = 8mod 9 F ρ = 8 8 + = 5mod 9 F th The j star umber (so called because geometrically these umbers ca be arraged to represet hexagrams is deoted as s j ad is of the form = 6j j + sj for j =,, 3, So, s =, s = 3, s3 = 37, ad so o. It is easy to show that s = j s + + j j for j =,, 3, Pictorially, s ca be represeted as ad s as depictig the six-corered star shape. ρ s =, ρ s = 4, ρ s =. I fact, Clearly, 3 Lemma 4.. The digital root of a star umber is always or 4. I fact, the progressio of digital roots of star umbers is, 4,,, 4,, Proof. Sice the digital root of ay iteger is oe of,,, 9, the digital root of a product of the form j( j is 9,,6,3,,3,6,,or 9 (0 represeted as 9. Cosequetly, the digital root of a product of the form 6j j is oe of 9,3,9,9,3,9,9,3,or 9. Hece the digital roots of star umbers are, 4,,, 4,, 300
5. A Applicatio Here is a problem simple problem. Prove that T = + + + 50 is divisible by 9. Here we will apply Propositio.. We write ( ( ( ( T = 50 + 49 + 48 + 47 + 46 + 45 + + 8 + 7 + 6 + 5 + 4 + 3 + + 3 3 = 50 + 49 + 48 + 47 + 46 + 45 + + 8 + 7 + 6 + 5 + 4 + 3 + 9 But by Propositio. the sum i each parethesis is divisible by 9, ad hece so is their sum, ad their sum plus 9. Here is aother problem that ca be solved usig digital roots. Problems similar to this oe ca be foud i Polya (957 [9] ad (Noller, et al. 978 [0]. Suppose we have a five-digit umber. We are give that this umber is divisible by 7. Startig with the first oe, how may digits of this umber must be disclosed before we ca uiquely determie it? Assume we are give the first digit, say 4. Obviously, more iformatio will be eeded before a uique solutio is foud. For example, 46,800 = 650 7, 48, 600 = 675 7, all fit the bill. So, assume ow the secod digit is also give, say 8. Agai, we caot fid a uique solutio based o this iformatio: 48, 3 = 67 7, 48, 600 = 675 7, are all possible solutios. So, assume oe more digit is give, say 9. We claim this would be eough to solve the problem. If a umber is divisible by 7, it must be divisible by both 8 ad 9. But a umber is divisible by 8 oly if oe of the two coditios holds: The hudreds digit is eve ad the last two digits are a multiple of 8 or the hudreds digit is odd ad the last two digits are a multiple of 4 but ot 8. Sice i our example the hudreds digit is odd, the last two digits of the umber we are looig for must be a multiple of 4 but ot 8, that is, the last two digits must be oe of 04 0 8 36 44 5 60 68 76 84 9 O the other had, to be divisible by 9, the digital root of the umber must be 9. Sice ρ 48, 904 = 7, ρ 48, 9 = 6, ρ 48, 90 = 5, ρ 48, 98 = 4, ρ 48, 936 = 3, ρ 48, 944 =, ρ 48, 95 =, ρ 48, 960 = 9, ρ 48, 968 = 8, ρ 48, 976 = 7, ρ 48, 984 = 6, ρ 48, 99 = 5, we ow that the umber must be 48,960 = 680 7. Refereces [] O Beire, T.H. (96 Puzzles ad Paradoxes. New Scietist, No. 30, 53-54 [] Garder, M. (987 The Secod Scietific America Boo of Puzzles ad Diversios. Uiversity of Chicago Press, Chicago. [3] Trott, M. (004 The Mahematica Guide Boo for Programmig. Spriger-Verlag, New Yor. http://dx.doi.org/0.007/978--449-8503-3 [4] Ghaam, T. (0 The Mystery of Numbers: Revealed through Their Digital Roots. d Editio, Create Space Publicatios, Seattle. [5] Dudley, U. (978 Elemetary Number Theory. Dover, New Yor. [6] Pritchard, C. (003 The Chagig Shape of Geometry: Celebratig a Cetury of Geometry ad Geometry Teachig. Cambridge Uiversity Press, Cambridge. [7] Hide, H.J. (974 The Additive Persistece of a Number. Joural of Recreatioal Mathematics, 7, 34-35. [8] Averbach, B. ad Ori, C. (000 Problem Solvig through Recreatioal Mathematics. Dover Publicatios, Mieola. [9] Polya, G. (957 How to Solve It: A New Aspect of Mathematical Method. d Editio, Priceto Uiversity Press, Priceto. [0] Noller, R.B., Ruth, E.H. ad David, A.B. (978 Creative Problem Solvig i Mathematics. State Uiversity College at Buffalo, Buffalo. 30