Permuted Multiples

It can be seen that the number, $125874$, and its double, $251748$, contain exactly the same digits, but in a different order.
Find the smallest positive integer, $x$, such that $2x$, $3x$, $4x$, $5x$, and $6x$, contain the same digits.

This type of problem can be usually solved by writing a program and it’s typically hard to solve it manually. Anyway, I will provide an explanation and the solution.

For the smallest x where 2x, 3x, 4x, 5x, and 6x contain the same digits, you need to discover a number where this requirement fulfils.

Here’s a logical way:
You can begin by assuming that ‘x’ is a 1-digit number. But, obviously, a 1-digit number multiplied by 2, 3, 4, 5, 6 would always result in different digits. Hence, our assumption is wrong.

Next, let’s move forward by considering ‘x’ as a 2-digit number. Multiplying a 2-digit number by 2, 3, 4, 5, 6 would result in 2-digit numbers or 3-digit numbers. Hence, ‘x’ can’t be a 2-digit number.

The next logical step would be to consider ‘x’ as a 3-digit number. However, a 3-digit number multiplied by 6 (the highest multiplier here) would result in a 4-digit number. So, ‘x’ can’t be a 3-digit number either.

So, we realize that ‘x’ must be of form 1xx…xx2, where ‘x’ can be any digit. This helps us narrow down the possibilities significantly. Now, we try and check using computer code iterating from 100000 which by multiplying by 6 is still a six-digit number.

Upon iterating, we find that the smallest number which satisfies this condition is $142857$. This is a special number known as a cyclic number. Upon multiple of {2, 3, 4, 5, 6}, we get:

$2x = 285714$

$3x = 428571$

$4x = 571428$

$5x = 714285$

$6x = 857142$

Each of these five results contains the exact same digits as $142857$, just in a different order. Therefore, $142857$ is the smallest positive integer that fulfills these criteria.

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