Proof of divisibility Test

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 divisibil...