A divisibility test for 19

Seven: Divide the number into threes, then alternately subtract and add them till you get a smaller number that you know is or isn't divisible by 7. For example 1298311®1-298+311 = 14 = 2·7 so it's divisible by 7.

Eleven: Divide the number into twos, then add them, e.g. 1824834® 1+82+48+34 = 165®1+65 = 66 = 6·11 so it's a multiple of 11. (see Issue 2.2 for a different test)

Thirteen: Divide the number into threes then alternately add and subtract, but make sure the first part is multiplied by -1, e.g. 16878043®-16+878-043 = 819 = 63·13 so it's divisible by 13.

Having told us that these tests depend on the fact that 9×11 = 99 and 7×11×13 = 1001, Mr Greenacre asks us if we know of any for 17, 19, 23,  etc. In reply to this, we can only offer the following test for 19, which was discovered recently in different parts of the world: by Apoorva Khare, a Sixth Form student in Orissa, India, and independently by Murray Humphreys of the Maths Dept at Jomo Kenyatta University and Nicholas Macharia of Karatina Children's Home in Kenya.

Suppose we have a number 12432099. Remove the unit digit, double it, and add it to the numbers remaining to get 1243209 + 2·9 = 1243227. Repeat the process - 124322+14 = 124336 - again and again - 12445,  1254,  133,  19  until you get a number you know is (or isn't) divisible by 19. Here the answer is yes.

Can our readers PLEASE help us fully answer Greenacre's question?!

References:

1. "Divisibility Tests" by A. Khare, Furman University Electronic Journal of Undergraduate Mathematics, Volume 3, 1997, pp 1-5.
2. "Tests for divisibility by 19" by M. Humphreys and N. Macharia, The Mathematical Gazette, Vol 82, No. 495, November 1998, pp 475-477.

The first journal is a free web-based journal which welcomes mathematical research (not just original work, but expository and survey articles as well) from school or undergraduate students. The second is a (not free, but very very good) publication of the UK Mathematical Association.