52 European Journal of Operational Research 42 (1989) 52-58 North-Holland

Theory and Methodology

Two suggestions to improve on the efficiency of the check computations in the banking system in Belgium

P a t r i c k S T E V E N S Brussels Free University, Faculty of Applied Sciences, Department of Mathematical Analysis, Pleinlaan 2, 1050 Brussels, Belgium

Abstrac t : Choos ing m = 97 as modu lus for the c o m p u t a t i o n of the check digi ts in an account n u m b e r of a Belgian bank account , it was assumed that m had to be selected f rom the set of pr imes. We show that this restr ict ive a s sumpt ion is incorrect by in t roduc t ing m = 93 as a more eff icient modu lus than the cur ren t m = 97. The advan tage of using m = 93 appears in the capac i ty of de tec t ing two a l tered digits. W e also presen t a sl ight modi f i ca t ion of the check system, yet more reducing the p r o b a b i l i t y of an unde tec tab le e r ror pa t te rn . The p a p e r m a y be of prac t ica l interest for bank ings in those count r ies where a na t iona l s t a n d a r d s t ructure of account numbers does not yet exist, bu t where its e s t ab l i shment is given ser ious cons ide ra t ion to.

Keywords: Banking, cont ro l

1. Introduction

The cur ren t s t ructure of account numbers of bank accounts in Belgium dates back f rom 1969, when the assoc ia t ion of banks dec ided to un i fo rm these numbers at a na t iona l level. Each n u m b e r hence consists of twelve digits,

a 9 a s a 7 - a 6 a s a 4 a 3 a 2 a l a o - ClC o.

Throughou t this paper , we deno te

9 1

N = ~a i lO ' , c = ~ ce l0e . i = 0 p = 0

The check digi ts c 1 and c o are c o m p u t e d by reduc ing N m o d u l o 97, i.e. N = c (mod 97), 0 ~< c ~< 96.

Received December 1987; revised May 1988

The pu rpose of the check digi ts is to de tec t errors which may be caused by h u m a n in ter - ference in the bank t ransact ions , such as reading, wri t ing or copy ing the account numbers . W h e n j digi ts are al tered, we say that an er ror pa t t e rn of weight j has occurred. The mos t l ikely event is the occurrence of no errors at all in a number .

In o rder to measure the eff iciency of a check system, it is useful to d i spose of a f requency d i s t r ibu t ion of nonzero error pa t te rns . Refe r r ing to some specific inves t igat ions p resen ted in [3-5], and no o ther concre te resul ts be ing avai lable , we assume that the fol lowing s ta tements a re signifi- cant : • E r ro r pa t t e rns of weight 1 have a p r o b a b i l i t y of a lmos t 90%. • Er ror pa t t e rns of weight 2 have a p robab i l i t y of 10%, subdivis ib le in 7% p r o b a b i l i t y of in te rchang-

0377-2217789753.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)

P. Stevens / About the control system in bank account numbers 53

ing two digi ts and 3% p robab i l i t y of a l ter ing ( though not in terchanging) two digits. • Er ror pa t t e rns of weight 3 or more have a p r o b a b i l i t y of less than 1%.

Because of their d o m i n a n t f requency within the class of nonzero er ror pa t terns , it is obvious that er ror pa t t e rns of weight 1 must abso lu te ly be detec ted .

I t is des i rable that er ror pa t t e rns of weight 2 are de tec ted as much as possible; this will be the cr i te r ion to d is t inguish be tween dis t inct check sys- tems.

Er ror pa t te rns of weight 3 are genera l ly cal led d is tor t ions . I t is ha rd ly feasible to s tudy then sys temat ica l ly and their occurrence is so much unl ikely that we will no longer take them into cons idera t ion .

Inqu i r ing af ter the motives of using m = 97 as check modulus , the p rope r inst i tut ions, such as banks and Publ ic Records Office, r epor t the fol- lowing arguments : • W h e n it was agreed to use a two-digi t modulus , m = 97 was chosen being the largest p r i m e less than 100, in o rde r to avoid zero-divisors in con- gruences. • The most f requent er ror pa t terns , namely those of weight 1 and the in terchanges of weight 2, are de tec ted by the m o d u l o 97 check system.

W e conc lude that nei ther n o n p r i m e modu l i nor a l t e ra t ions of weight 2 were ever involved in the concep t ion of the current check system. F o r this reasons, we argue in favour of an or iginal and comple t e s tudy, cons ider ing m a n y integers as po ten t i a l modul i and examin ing their capaci t ies of de tec t ing as many as poss ib le error pa t t e rns of weight 1 and 2 in general .

2. Choice of the modulus: Preliminary selection

2.1. I n t r o d u c t i o n

W e will hencefor th pay a t t en t ion to the single and doub le e r ror -de tec t ing capac i ty of any integer m: 70 < m < 100 as po ten t i a l check modulus . The fac tor iza t ion into p r imes of these integers is given in Tab le 1.

Us ing the same no ta t ions as in Sect ion 1, we s tate that an account number consists of integers N a n d c s u c h t h a t N = c (mod m ) , 0 ~ < c ~ < m - 1 . As errors m a y poss ib ly co r rup t the number , we

Table 1 Factorization into primes of potential moduli m

71 8 1 : = 34 91 = 7 . 1 3

72 = 2 3 . 3 2 82 = 2 - 4 1 92 = 2 2 . 2 3

73 83 93 = 3 - 3 1

7 4 = 2 . 3 7 84 = 2 2 . 3 - 7 9 4 = 2 - 4 7

75 = 3 - 5 2 85 = 5 . 1 7 95 = 5 - 1 9

76 = 2 2 - 1 9 86 -- 2 . 4 3 96 = 2~ .3

77 = 7 - 1 1 8 7 = 3 . 2 9 97

78 = 2 - 3 - 1 3 88 = 2 3 . 1 1 98 = 2 . 7 2

79 89 99 = 32 -11

80 = 2 4 " 5 90 = 2 " 3 2 " 5

denote its received version at the input of the check mode by

t t t t t t t t t ! t t

a g a 8 a 7 - a 6 a 5 a 4 a 3 a 2 a l a 0 - c 1 c 0 ,

set t ing

9 1

N ' = E a ; 1 0 ' , c ' = E c; 10p- i - - 0 p = 0

It is clear that an unknow n er ror pa t t e rn re- ma ins unde tec ted when N - N ' = c - c ' (mod m): the less solut ions this congruence admi ts , the bet- ter check modu lus m is. Therefore , we will discuss this p rope r congruence in the d i f ferent cases of the concerned error pa t terns .

2 .2 S i n g l e e r r o r

(1) O n e e r r o r in d ig i t a i (0 ~ i <~ 9). Then

N - N ' = ( a s - a ; ) 1 0 i = a l 0 i,

t s e t t i n g a = a i - a i; t h u s l y < Ic~[ ~<9. The error remains unde tec t ed when a l 0 i = 0

(mod m), be ing equivalent to m lodO i, where " x l Y" means " x is a d ivisor of y " , which means m = y2r5 s with y [ a, thus 1 ~ y ~< 9, 0 ~ r, s ~ 9.

On the basis of Tab le 1, we observe that the fol lowing integers mus t u n d o u b t e d l y be rejected as po ten t ia l check modul i : 72, 75, 80, 90, 96.

(2) O n e e r r o r in d i g i t Cp (0 <~ p <~ 1). Then

c - c ' = i l l 0 p ,

where

t f l = e p - C p , l ~ < [ f l [ ~ < 9 .

The error is not de tec ted when f l lO p = 0 (rood m).

I t is easy to see tha t this case is inc luded in the former.

54 P. Stevens / About the control system in bank account numbers

T a b l e 2

F a c t o r i z a t i o n i n t o p r i m e s o f 10 k - 1, 1 ~< k ~< 9

101 - 1 = 3 2

1 0 2 - 1 = 3 2 . 1 l

1 0 3 - - 1 = 33. 37

10 4 - 1 = 32 .11 • 101

1 0 5 - 1 = 3 2 - 4 1 - 2 7 1

1 0 6 - 1 = 3 3 . 7 . 1 1 . 1 3 - 3 7

1 0 7 - 1 = 3 2 . 2 3 9 . 4 6 4 9

1 0 8 - 1 = 3 2 . 1 1 . 7 3 . 1 0 1 . 1 3 7

109 -- 1 = 3 4 . 3 7 . 3 3 3 6 6 7

2.3. Double error

(1) In terchanging digits a i and aj (0 <~ i, j <~

9, i v~j). Then

N -- N ' = a i l O i + aglO j - ( a g l O i + ailO j )

= ( a i - a j ) (10 i - 10 y)

= a l0J (10 k -- 1),

where a = ai - a j , j < i is assumed, k = i - j . The error is not detected when al0g(10 k - 1) =

0 (mod m), with 1 ~< la I ~<9, 0 ~ < j ~ 8, 1 ~ < k ~ 9. This is equivalent to: m = y 2 r 5 s t ~ , where 1 ~<

~<9; 0~<r, s~<8; ~ ] 1 0 k - 1 , ~ < 100. On the basis of Tables 1 and 2, we omit the

following integers: 73, 74, 77, 78, 81, 82, 84, 88, 91, 98, 99. The remaining set of potential moduli is

J r ' " = {71, 76, 79, 83, 85, 86, 87, 89, 92, 93,

94, 95, 97}.

(2) In terchanging digits c o and c 1. It is easy to see that this case is included in the former.

(3) In terchanging digits a i and c o (0 <~ i <~ 9, 0 ~< p ~< 1). Then

U -- N " = ( a ~ - c e ) l O i, c - c ' = ( c e - - a~)10 p

T a b l e 3

F a c t o r i z a t i o n i n t o p r i m e s o f 10 ° + 1, 1 ~ v ~< 9

10 a + 1 = 1 1

102 + 1 = 101

103 + 1 = 7 - 1 1 . 1 3

1 0 4 + 1 = 7 3 - 1 3 7

105 + 1 = 1 1 . 9 0 9 1

106 + 1 = 1 0 1 - 9 9 0 1

107 + 1 = 11- 909091

10 s + 1 = 17 . 5882353

109 + 1 = 7 . 1 1 - 1 3 - 1 9 . 5 2 5 7 9

The error is not detected when fl(10 i + l0 p) = 0 (mod m), with fl = a i - cp, 1 ~< I/31 ~< 9.

We write this congruence as: fll0U(10 v + 1) = 0 (mod m), where 0 ~ < u ~ l ; 0 ~ < v ~ 9 ; (u, v) 4~ (1, 9).

It is equivalent to: m = ,{2r5s~, where I ~< y ~< 9; 0~<r, s~< l ; ~ [ 1 0 v + 1, ~ < 100.

On the basis of Tables 1 and 3, we delete the integers 76, 85 and 94 f rom ~ "

The remaining set is

J [ ' = {71, 79, 83, 86, 87, 89, 92, 93, 94, 97}.

(4) Alter ing digits a i and aj (0 <~ i, j <~ 9, i ~ j ) .

Then

N - - N ' = ( a i - - a ' i ) l O i + ( a j - - a j ) 1 0 ' J;

set k = a i - a ; , l = a g - a j . The error is not detected when k l 0 ~ + l l0 j = 0

(mod m), with 1 ~< [k I, I I[~< 9. This congruence will be thoroughly examined in the next section.

Grounded on an estimate of the theoretical probabil i ty P,,.th that an error pat tern of this type is not detected by the modulo m control, we may delete the integers 86, 92 and 94 f rom the set J r ' . The proof is given in Appendix B. The remaining set of potential moduli is

. / g = {71, 79, 83, 87, 89, 93, 97}.

(5) Alter ing digits c o and c 1. It is easy to see that this case is included in the former.

(6) Alter ing digits a~ and Cp (0 <~ i <~ 9, 0 <~ p <~

1). Then

N - N ' = ( a i - - a ; ) 1 0 i = a l0 ' ,

where a = a i - a ~ , l ~ < 11[~<9.

c - c ' = ( c p - c )10o = / 3 1 0 p ,

t w h e r e / 3 = c e - c o , l ~ < 1/31 ~<9.

The error is not detected when a l 0 ~ = f l l 0 p (mod m)

If i :g p, this case is included in (4). If i = p, the congruence is equivalent to: a = fl

(mod m), as every integer m ~.//t' is copr ime to 10.

i t Hence, when a o - a o = c e - c e (O<~p<~l) , an

error pat tern has occurred which cannot be de- tected by any modulus m. This is an inherent weakness of the check mode. In Section 4, we will show how this deficiency can possibly be avoided by slightly modifying the check system.

P. Stevens / About the control system in bank account numbers 55

W e conclude that, af ter a p re l imina ry selection, the set of po ten t ia l modu l i is ,//t '= {71, 79, 83, 87, 89, 93, 97}. W e have observed that they are

all capab le of detecing one error and two errors by in te rchanging two digits. In o rde r to select an op t ima l modulus , we will compare their capaci t ies of de tec t ing two a l tered digits.

3. Choice of the modulus: Definitive selection

We not ice that P93 < P97. A l though the ad- vantage of a p p r o x i m a t e l y l%v is small , it leads us

nevertheless to the s t a tement that m = 93 is a be t te r modu lus than m = 97, re la ted to the check system in Belgian banking . F r o m a prac t ica l po in t of view, however, the advan tage is too insignif i- cant to argue in favour of an a d a p t a t i o n of the current system, which has been set t l ing itself for a lmost two decades.

F r o m the foregoing selection, it results that we have to conside/" the congruence: k l 0 ~ + / 1 0 j = 0 (mod r n ) w i t h l ~ < I k [ , I l l ~<9 ;0~<i , j ~ < 9 ; i ~ j ; m ~ J / Z .

Wi thou t loss of genera l i ty we assume that i > j , / > 0 .

In A p p e n d i x B it is p roven that Vm ~,~¢: P,~.th = 1 / ( m - 1). We in t roduce however some o ther es t imate P,, of the p robab i l i t y that two a l te ra t ions are not de tec ted by the m o d u l o m control . I t must be clear that P~ offers a more realist ic a p p r o a c h

than Pm,th"

Definition. P,, = T,,,/n, where T m is the number of so lu t ions of the congruences : k l 0 i + 110 j = 0 (mod m ) , w h e r e l ~< [ k [ , l ~ < 9 , 0 ~ < j < i ~ < 9 , m ~ J [ .

n = 18 • 9 • 45, be ing the p roduc t of the number of poss ibi l i t ies for respect ively k, l and the couple (i , j ) .

The results of a compu te r p r o g r a m are pre- sented in Tab le 4.

F o r the p r inc ipa l cases m = 93 and 97, the so lu t ions are expl ic i ty l is ted in A p p e n d i x A. In o rde r to count T,,, we observe that Vm ~ . / g , the congruence k l 0 i + ! = 0 (mod m) is equivalent to the 10 - i congruences k l 0 "+' + / 1 0 " = 0 (mod m), O ~ u ~ 9 - i .

F o r an extensive t r ea tment of congruences, the reader is referred to [1,2].

T a b l e 4

m T m Pm (%) P~,th (%) Pm//Pm,th

71 81 1.111 1.429 0.78

79 89 1.221 1.282 0.95

83 81 1.111 1.220 0.91

87 87 1.193 1.163 1.03

89 67 0 .919 1.136 0.81

93 57 0 .782 1.087 0.72

97 64 0 .878 1.042 0.84

4. Additional improvement on the check system

4.1. Introduction

The idea this sect ion conta ins , was b rough t to my a t t en t ion by Professor L. Van H a m m e of Brussels F ree Univers i ty .

We will s tudy in deta i l the e r ro r -de tec t ion capaci t ies of modul i m = 93 and m = 97 in rela- t ion to the a l te ra t ion of two digits. In the previous section, we made the fol lowing subdiv is ion : • Al te r ing digi ts a i and aj (0 <~j < i <~ 9): 45 ins tances are inc luded; we a l ready not iced that

T93 = 57 and T97 = 64. • Al te r ing digits c o and c1: the co r r e spond ing congruence yields 1 so lu t ion for bo th values of m. • Al te r ing digi ts a, and cp (0 <~ i <~ 9, 0 <~p <~ 1):

i :~p: the 18 co r r e spond ing congruences yield 24 solut ions for bo th values of m (see A p p e n d i x A),

i = p: the 2 co r re spond ing congruences yield 18 solut ions for bo th values of m.

Def in ing T£ as the to ta l n u m b e r of so lu t ions of the congruences , associa ted with the 66 poss ib i l i - ties to choose two pos i t ions where the a l t e ra t ions

occur, we ob ta in T9' 3 = 100, 7"97 = 107. W e suggest the fol lowing mod i f i ca t ion of the

check system: to the 10-digit n u m b e r N, a 2-digi t check number b is a d d e d such that the ent i re 12-digit n u m b e r B = 100N + b is a mul t ip le of the check modu lus m, i.e.

11

B = Y" bilO i= lOON + b = O ( m o d m ) . i = 0

Alte r ing two digits b i and bj (0 ~<j < i ~< 11), t t and set t ing k = b i - b i , l = b j - b j , such an error

pa t t e rn is not de tec ted when klO' + 110 j = 0 (mod m). W e def ine T£' as the n u m b e r of so lu t ions of these 66 congruences , where 1 -%< I k [, l ~< 9; 0 ~<j < i ~ < 1 1 .

56 P. Stevens / About the control system in bank account numbers

By means of Appendix A, it is easily counted t ! I t

t h a t T93 = 8 4 ; T97 = 9 7 .

4.2. Observations

(a) T~' < T,~, implying that the modified system is more efficient than the current system;

(b) Tg~' < T97', meaning than m = 93 is again a better modulus than m = 97 to be used in the modif ied system;

(c) T97' - T93' > T9~ - T9' 3, showing that the ad- vantage of using m = 93 instead of m = 97 is even greater in the modif ied system.

4.3. Conclusion

We assert that error detection in Belgian bank- account numbers would prove to be opt imal when N and b (0 ~< b ~< 92) would fulfil the condit ion that 100N + b = 0 (mod 93).

Otherwise stated: an account number, consid- ered as a single 12-digit integer, should be a multiple of m = 93.

T a b l e 5

i m = 93 m = 97

k l k 1

1 9 3

2 - 1 7

3 - 4 1

- 8 2

4 - - 2 5

5 - - 4 7

6 - - 3 6

7 - 8 5

9 6

8 - - 5 8

6 9

9 - - 6 3

10 - - 7 4

11 - 5 2

9 7

- 1 3

- 2 6

- 3 9

3 7

- 1 9

1 7

7 5

- 9 5

5 8

- 6 1

- 3 5

- - 2 1

- 4 2

- 6 3 - - 8 4

- 1 5

5. Practical importance

A uniform structure of bank-account numbers, such as it exists in Belgium, has not yet been established in most other countries till now. It is however very likely that they are considering to do so, regarding the interesting applications that be- come feasible, such as fluent transactions between distinct banks and s tandardized electronic bank- ing. It is worth while noticing that even the two competi t ive automat ic payment systems in Bel- gium have recently merged into one national sys- tem. Moreover, the intended s tandardizat ion within the European C o m m u n i t y towards 1992, is a recent argument that reveals the interest of this paper. Related to banking, it is imaginable that all EC members will agree upon applying the same structure for their bank-account numbers, in order to facilitate financial t ransactions on an interna- tional level.

The aim of this paper is to contr ibute to the establishment of a check system with max imum error detection capacity, possibly founded upon the Belgian system, though not in its current fea- ture.

Note. W.J. Van Gils of Philips Research Labora- tories, Eindhoven, The Netherlands, observed quite rightly that every solution (k, l) of a congruence k l0 i + 110 j = 0 (mod m) must actually be weighed by /x (k , l) = (10 - I k 1)" (10 - 1)/100, in order to obta in a tighter approach of the probabil i ty that two altered digits are not detected. Indeed, the funct ion bt(k, 1) equals the probabil i ty that two alterations with values k and I can simultaneously corrupt the digits ai and aj. As an example we consider k--- - 2 and 1 - -6 ; an error with value k = - 2 at digit a~ can occur iff a~ ~ (9, 8, 7, 6, 5, 4, 3, 2) and an error with value l = 6 at digit aj can occur iff aj E (0, 1, 2, 3}. Hence the probabil- ity that two alterations with values k and 1 can simultaneously corrupt the digits a~ and a j, is 3 2 / 1 0 0 = /x (k , l).

We have recomputed our above-ment ioned numerical results by applying the weight funct ion /~(k, l) to the solutions of the congruences (see Table 5) and we have noticed that the final con- clusions remain unchanged.

P. Stevens / About the control system in bank account numbers 57

Appendix A

Table 5 con ta ins the solu t ions of the con- gruence k l 0 ~ + l = 0 (mod m) with 1 -%< I k I, l-%< 9; 1~<i~<11; m - 93, 97.

Appendix B

We will formular ize Pm,th, def ined as the theo- ret ical p robab i l i t y that the modu lo m-cont ro l does not detect two a l tered digits. Therefore , we ex- amine the congruences k l 0 ~ + 110 j = 0 (mod m), with 1 ~ I k l , l ~ < 9 ; 0 ~ < j < i ~ < 9 .

Each t ime we assume that i, j and l are f ixed and we invest igate the p robab i l i t y that a so lu t ion k exists such that 1 ~< I k ] ~< 9. Gene ra l theorems on the solvabi l i ty of l inear congruences can be found in references [1] and [2].

We are in teres ted in three subcases: • m is copr ime to 10, for example m ~ J t ' . A n equivalent def in i t ion is: g c d ( m , 1 0 ) = 1, where " g c d ( a , b ) " denotes the greates t c o m m o n divisor of integers a and b.

• m = 2 m ' and gcd (m ' ,10 ) = 1, for example m = 86, 94.

• m = 4 m " and g c d ( m " , 1 0 ) = 1, for example m = 92.

1. g c d ( m , 10) = 1

The congruence: k l 0 ~ + 110 j = 0 (mod m) is equiva lent to the congruence: k l 0 ` - j + l = 0 (mod m). F o r f ixed values of i, j and l 4: 0, this con- gruence has one solut ion k: 0 < k < m. The p r o b - ab i l i ty that 1 ~< I k I ~< 9 is 1 8 / ( m - 1). The p rob - ab i l i ty that this very solut ion also equals the com-

I

mit ted error k = a i - a , is ~ . W e conc lude that

Pm,th = 1 / ( m -- 1).

2. m = 2 m ' and g c d ( m ' , 10) = 1

2.2. 1 ~<j < i -%< 9 (36 instances) 10 'k + 10Jl = 0 (mod m) if and only if

10 i 10 j k + ~ - l = 0 ( m o d m ' )

has one solut ion: 0 < k < m ' . Then

36 1 P 2 2 = 4-5" m ' - I

Hence we f ind that

8 1 16 1 P~'th = 9 " m ' - - I = - 9 - " r n - - 2

3. m = 2 m ' = 4 m " and g c d ( m " , 10) = 1

3.1. 2 -%<j < i -%< 9 (28 instances)

10 'k + 10q = 0 (mod m) if and only if

10i 10J m " ) --4--k + --~-- l = 0 ( m o d

has one solut ion: 0 < k < m " . Then

28 1 P31 = 4-5 " r n " - 1 "

3.2. j = 0, i = 1 (1 instance)

10k + l = 0 (mod m) is so lvable if and only if gcd(m, 10) 1l. Thus l = 2 l ' if and on ly if 5k + l ' - 0 (mod m ' ) has one solut ion: 0 < k < m ' . Then

1 4 1 P 3 . : = 4--5 " 9 " m ' - 1 "

3. 3. j = 0, 2 -% i -%< 9 (8 instances ) 10'k + l = 0 (mod m) is solvable if and only if

gcd (m, 10 ' ) [l. Thus l = 4 l " if and only if 1 0 ' / 4 k + l " = 0 (mod m " ) has one solut ion: 0 < k < m " . Then

8 2 1 P3.3= 4 ~ ' 9 " m " - I

2.1. j = 0, 1 -%< i ~< 9 (9 instances)

lO 'k + l = 0 (mod m) is solvable if and only if gcd (m, 10 i) 1l, thus l = 2 l ' if and only if 1 0 i / 2 k + l ' = 0 (mod m ' ) has one solut ion: 0 < k < m ' .

The con t r ibu t ion to P,,,th appa ren t l y is

9 4 1 P 2 . , = 4-5" 9 " m ' - l "

3.4. j = 1, 2 -%< i -%< 9 (8 instances)

10 'k + 10l = 0 (mod m) is so lvable if and only if gcd (m, 10 i) 110l. Thus l = 2 l ' if and only if l O i / 4 k + 5 l ' = 0 (mod m " ) has one solut ion: 0 < k < m" . Then

8 4 1 P 3 4 = 4 ~ ' 9 " m " - 1

58 P. Stevens / About the control system in bank account numbers

Hence we obtain

1 [ l ~] 1 4] Pm'th--45[m'----rr-L~--I [28+196+ + - - ' m ' - - I -9]

1 [ 1 0 0 . . . . 1 4 ) ] =4--5 3 m " - I + 9 m - 1 "

computat ion of T,, for these integers and compar- ing to the values in Table 4.

We obtain: T86 = 137, Tgz = 242, T94 = 137. A general necessary condition for an integer rn

to be a valuable modulus is gcd(m, 10) = 1.

Conclusion References

For a modulus m coprime to 10, we have approximately P,,,th = 1/m.

For m = 2 m " and gcd(m' , 10)= 1, we have Pm,th --~ 2 /m.

For m = 4 m " and gcd(m", 10)= 1, we have Pm,th ----- 3 / m .

These results justify the omission of the in- tegers 86, 92 and 94 as potential moduli, as we did in Section 2, (4). Besides, this is confirmed by the

[1] Agnew, J., Explorations in number theory, Brooks/Cole, Monterey, CA, 1972.

[2] LeVeque, W.J., Fundamentals of number theory, Addison- Wesley, Reading, MA, 1977.

[3] Uitert, C. van, "Codestelsels en controle technieken", In- formatie 13/5 (1971).

[4] Verhoeff, J., "Wiskundige aspecten van het Nederlandse administratienummer voor personen", lnformatie 12/4 (1970) 162-169.

[5] Zwolsman, P.A.M., "Controlecijfers: Enige methodo- logische aspecten", Informatie 17/1 (1975) 28-36.