Tuesday, March 7, 2017

elementary number theory - A divisibility rule for 19


Proof the following divisibility test for 19:


Add two times the last digit to the remaining leading truncated number. If the result is divisible by 19, then so was the first number.



More mathematically:


Let a,bZ. Proof that 10a+b is divisible by 19 if a+2b is divisible by 19.


My guess is that we can proof this using congruences.


Answer



10a+b is divisible by 19 if and only if 20a+2b is divisible by 19, of course 20a+2ba+2bmod


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