November 2022

Mathematical Matrix

On Two Interesting Properties of Primes, p, with Reciprocals in Base 10 having Maximum Period p-1.

Inverses of primes which have maximum period property, namely, inverses of primes, p, whose decimal representation repeats after p-1 are well studied in the mathematical literature. Examples of such primes are 7 and 19 with,

1/7 = 0.142857142857…. or 1/19 = 0.052631578947368421 and so on.

1/11 or 1/13 don’t have such property as it is easy to check.

1/11 =.0909090909.

1/13 = 0.076923,076923 …

The Ekidhikena Purvena rule (by one more than the one before) from Bharati Krishna Tirtha’s Vedic Mathematics works beautifully for finding decimal digits of inverse of numbers which ends with 9.

For example, 1/19 =0.05263157894736842105263…. We start with 1 which is the last digit of recurring period, 052631578947368421 and multiply it by 2 repeatedly. When the result of multiplication is more than one digit, we carry it and add it to the result of next multiplication by 2 and so on. The process is shown below.

1 0 1 0 0 1 1 1 1 0 1 0 1 1 0 0 0 0

0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 x 2

The result also can be obtained by dividing 1 by 2 and using the remainder in the next division as shown below.

1/2     052631578947368421

            101001111010110000

Note that when we divide 1 by 2, we get 0 and remainder 1. Next time we use 1 and 0 as 10 and divide it by 2 to get 5. This process repeats.

Such methods of inverting numbers without the division operation is always desirable.

We have given a table to find inverses of primes such as 7,17,19,23,29,41, 59,61,97,109,113,131,149,167,179,181,193,223,229,233,257,263,269,313,337,367,379,383,

389,419,433,461,487,491,499,503,509,541,571,577,593,619,647,659,701,709,727,743,811,

821,823,857,863,887,937,941,953,971,977,983.

This includes all such primes up to 1000.

We do this by finding a pair of numbers a and b which can be used for finding decimal digits of inverse of a prime without division.

For example, decimal digits of 1/7 can be found using the pair 7 and 5 using the operation shown below. We call that operation extended multiplication of 7 by 5 to differentiate it from conventional multiplication rule used in arithmetic.

2 1 4 2 3 0

1 4 2 8 5 7 x 5

Note that one period is, 142857. We obtain it by starting with 7 and multiply it by 5 to get 35. Use 5 as the preceding digit and multiply it by 5 and add carry 3 to get 28. Use 8 as the preceding digit and keep 2 as carry.

Table 1 gives multiplication factor to be used and the starting digit for all such primes up to 1000.

Table 1: Factors Needed for obtaining decimal digits of inverse of primes

 

Prime Number, p Last Digit of 1/p after which digits repeat (a) Multiplication factor (b)
7 7 5  
17 7 12  
19 1 2  
23 3 7  
29 1 3  
47 7 33  
59 1 6  
61 9 55  
97 7 68  
109 1 11  
113 3 34  
131 9 118  
149 1 15  
167 7 117  
179 1 18  
181 9 163  
193 3 58  
223 3 67  
229 1 23  
233 3 67  
257 7 180  
263 3 79  
269 1 27  
313 3 94  
337 7 236  
367 7 257  
379 1 38  
383 3 115  
389 1 39  
419 1 42  
433 3 130  
461 9 415  
487 7 341  
491 9 442  
499 1 50  
503 3 151  
509 1 51  
541 9 487  
571 9 514  
577 7 404  
593 3 178  
619 1 62  

 

647 7 453
659 1 66
701 9 631
709 1 71
727 7 509
743 3 223
811 9 730
821 9 739
823 3 247
857 7 600
863 3 259
887 7 621
937 7 656
941 9 847
953 3 286
971 9 874
977 7 684
983 3 295

 

Subramani. K

E-mail: subramani.k@iiitb.ac.in

For more details, contact Mr. Subramani K at subramani.k@iiitb.ac.in