This post is for the theory of osculation  in Vedic Mathematics for  testing divisibility of a number by a prime number also called as "Ekadika Test"
"Ekadika" means one more.
To get ekadhika of a number find a multiple of that number, which has a 9 as its last digit. add one to its remaining right side part. For example: for 7 ekadhika is, 7*7 =49; 4 +1 is 5; for 13 : 13*3 is 39; 3+1 "ekadika" is 4; For  27 , 27*7 =189;
ekadika is 19;
To test the divisibility of any numbers by a prime number you first find out its ekadika and osculate.
Test1 :find is 91 divisible by 7
ekadika of 7 is 5, now osculate : 9+5*1 = 14; divisible by 7;so is 91;
Check 78 divisible by 13:
ekadika is 4 ; so 8*4 +7 =39; divisible by 13;
so 78 is divisible by 13;
To see 247 divisible by 19?
ekadhika(E) is 2; 7*2+4 = 18;  18*2+2 =38 which is divisible by 19;
2    4  7
38  18
So the number 247 is divisible by 19;

Now we prove "ekadika " test for divisibility of prime numbers ; theory of osculation in #vedic #Mathematics
If a number T divisible by a prime number k; then ,there exists an m s.t
T = m*k; T,m,k is in I - Integers != 0,1  ;
Any integer can be  written as 10d +s ; where 's' is the last digit;
T = 10d +s;
we prove that iff;
d + s*E is divisible by k then
10d + s = T is divisible by k;
if d + s*E is divisible by k
then d+s*E = kn; for some n.where E is ekadhika , E,n are integers
E =( kp +1)/10 - ekadhika definition;
d + s* (kp+1)/10 = kn;(p <n);
10d +skp +s =10kn;
10d +s =10 kn - skp;
T = (10n - sp)*k
T = m*k implies T is divisible by k; m = 10n -sp which is not 0 ,1 since 10 >s and p <n ;
hence the proof;
Something for  Chika on his next summer Holidays #chikastest