Problems & Puzzles: Puzzles

Puzzle 188. Dividing large numbers

Now a very simple task:
 

What is the smallest number S divided by the prime 692399, being S the concatenation of the first K natural numbers 1234....K?

Solution:

Ken Wilke, Jim Howell and Flavio Torasso found - each one by his own side - the solution asked:

692399 divides to the number S = 1234...K, for K= 8,468,396 having S 58,167,668 digits for this K value.

Each one discovered by their own that solving this puzzle needs not - by any means - a big numbers package software, but only a very small and simple code to emulate the basic algorithm of the division "by hand". The basic feature of this algorithm is that you need not consider the number S in all his extension but only one digit at a time - form left to right - that you will concatenate to the residue of the previous step, etc...

The three solvers also found several other K values larger than 8,468,396 such that S is again divided by 692399.

In particular Jim found the first instance where K is also a prime number: K = 21830773, having S 163,535,081 digits for this K value.

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As a curios note I would like to add that if we relax the condition of taking all the digits of the final K number to just taking some of the first digits of it, then the smallest solution is this one:

692399 divides to the number S = 1234...K, for K= 7342, taking only the first digit of K, having S 28,258 digits.

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