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,b∈Z. 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+2b≡a+2bmod19
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