For example, to test div. by 13, multiply the last digit by 4 and add to the priors. So consider 26. 6x4=24. 24+2=26. So you just have to know 26=2x13. Let's try 1599. 4x9=36. 36+159=195. 4x5=20. 20+19=39. 4x9=36. 36+3=39.
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I guess I also learned of the test for div by 19 on the other place b4 it became a cesspool. Multiply the last digit by 2 and add to the priors. Take for example, 1596. Double 6, and add 159+12=171. 1x2=2, add to 17. You get 19. For the 19s test the number always reduces to 19. Cool, huh?
The 19s test is similar to one for 7. Double the 1s place and subtr. from the priors. For example 8281. 1x2=2, 828-2=826. 82-12=70. 70 is divisible by 7. Wait! there's a test for 49: multi 1s dig by 5 and add to the priors. 828+5=833. 83+25=98. 9+40=49. FWIW, 8281=91x91. Go on, test for div by 13.
I'll quickly remind you the tests for div by 3,9 and 11. For 3 add all the digits, if the result is div by 3 then the number is div by 3. For 9 add the digits, if the result is div by 9, then the original number is div by 9. For 11 take the alternating sum of the digits analogous to an Euler Char.
The tests that I have in mind are all the same. Multiply the 1s place by a number that is smaller than 1/2 the prime whose div'ity being tested. Then add (or subtract) to the remaining digits. If the result is div by the prime in question, so is the original number. Here are a few more tests.
To test for divisibility by 23, multiply the 1s place by 7 and add to the priors. So consider 4899. 9x7=63. 489+63=552. Repeat process. 2x7=14, 55+14=69. If you try this again, then 7x9=63, 63+6=69. So you have to know 69=3x23. But also, 7 is 1 mod 3. So you're also testing for div by 3.
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