The number with no new digits

HardBrainteasers~8m

There is a $6$ -digit integer $N$ such that each of $2N, 3N, 4N, 5N, 6N$ uses exactly the same set of digits as $N$ (indeed each product is a rearrangement of $N$ 's digits). What is $N$ ?

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Think about the decimal expansion of a simple fraction whose digits cycle.
It's the repeating block of $1/7$ : $142857$ . Its multiples $2\times,\dots,6\times$ are rotations of it.

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142857

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Asked at: Jane Street, Citadel

The number with no new digits

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