Your only flaw here is that you've subjected yourself to rounding errors. 199/98 is not exactly equal to 2.03061224, and neither is 5/98 exactly equal to 0.05263158. As far as I can tell, there's no diophantine solutions to c = 2.03061224k + 0.05263158, but there

*is* a solution (in fact there are infinitely many such solutions) to c = 199/98k + 5/98. If you try your previous brute force method again with the corrected

equation, you'll see that c = 262, k = 129 is a solution.

The trick here is that we know 199/98k + 5/98 must be an integer. If we pull out the common factor of 1/98, we see that 1/98(199k + 5) must be an integer, and from that we can deduce that 199k + 5 must be a multiple of 98. Where do you think you'd go from here?

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