Henry Ernest Dudeney/Modern Puzzles/55 - The Repeated Quartette/Solution

Modern Puzzles by Henry Ernest Dudeney: $55$

If we multiply $64253$ by $365$ we get the product $23452345$, where the first $4$ figures are repeated.

What is the largest number that we can multiply by $365$ in order to produce a similar product of $8$ figures repeated in the same order?

There is no objection to a repetition of figures -- that is, the $4$ that are repeated need not be all different, as in the case shown.

$273863$

The key point is that a number of the form $\sqbrk {abcdabcd}_{10}$ is equal to $10001 \times \sqbrk {abcd}_{10}$.

We also have that:

$10001 = 73 \times 137$

and:

$365 = 5 \times 73$

So the $8$ digit number we end up with is a multiple of $5 \times 73 \times 137 = 50005$.

All we need to understand is that it has $2$ repeated sets of $4$ digits and ends in $5$.

The largest one of those is:

$99959995 = 273863 \times 365$

and the job is done.

$\blacksquare$

1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $55$. -- The Repeated Quartette
1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $106$. The Repeated Quartette