This doesn't prove the existence of such a number. It only gives the conditions for which the number must satisfy. Do we actually know such a number exists?
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Yep. N=1196+2500k k={0,1,2,3,4,...}
End of conversation
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Going one step further: The sixth-to-last digit is odd because only 166336 366336 566336 766336 966336 are divisible by 64
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