Numbers : Divisibility Osculation

finding the divisibility of a number by 7,13,17,19.the method is known as osculator method.first we have to know the concept of osculator.the osculator of 7 is 2 ( 7* 3 = 21 = 20+1 here we are adding 1 so we need to consider the osculator as negative osculator )

 similarly for 13 ,13 * 3 = 39 = 40-1 so the osculator is 40 and is one more osculator and the value is 4 and for 17 ,17 * 3 = 51 = 50+1 so again the osculator is negative and it is 5 and for 19 it is one more osculator  19 = 20-1 ,so the osculator is 2 .

if you didn't understand the concept of osculator just remember these

for 7 osculator = 2 , sign = ' - '

for 13 osculator = 4 , sign = ' + '

for 17 osculator = 5 , sign = ' - '

for 19 osculator = 2 , sign = ' + '

lets proceed with an example

55277838 is considered and to check it's divisibility by 7.

5527783 8  : 5527783 - 8 *2 =5527767 ( here we have taken the units digit and multiplied it by osculator 2 and subtracted it from the remaining part as the sign is negative as suggested above we have subtracted )

the next step is the following is repeated for the resulting value in the above step

552776 7   : 552776 - 7 *2 = 552762

55276 2 : 55276 - 2 * 2 = 55272

5527 2 : 5527 - 2 *2 = 5523

552 3 : 552 - 3 *2 = 546

54 6 : 54 - 6*2 = 42 as 42 is divisible by 7 so the number is divisible by 7.