Separate the number into groups of two digits as usual. Divide the first group of digits by two and add it to the tens of the next group. Take 373979. 37/2=18.5. 185+39=224. 224/2=112. 112+7=119, so we have 1199. 11/2=5.5. 55+99=154. 1/2=0.5. 5+54=59=2 (mod 19) so 373979=2 (mod 19). Note how we have previously shown that 373977 is divisible by 19.
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