I may be excused for not having Dudeney's tastes for amusements. So, no, I am not going to solve that problem.

One thing I suspect however is that both of you (Dudeney and you) can't be right. You got 5 fractions. Now the task is to find an integer such that any portion of it defined by any of the five fractions is an integer.

The four denominators 257, 809, 562, 795 are mutually prime, i.e., not two of them have common factors, except for 1. This means that the smallest number divisible by all 4 is their product: 92893449270.

If I were you, I would double check your calculations. The fractions should add up to 1, right? In your case it's rather 64319503951/92893449270.