Properties of 142,857

Theorem
This page gathers together some properties of $142 \, 857$ which arise through its being the digits of the recurring part of the reciprocal of $7$.

Multiplication of $142 \, 857$ by numbers higher than $7$ produces a similar pattern to when you multiply it by a single digit, but with added complications.

For example:

This becomes $714 \, 285$ when you take the $1$ off the front and add it to the back.

The exception is when you multiply it by $7$ or a multiple of $7$:

From Recurring Part of Fraction times Period gives 9-Repdigit, it is seen that this property is shared of all numbers formed from the digits of the recurring part of a recurring fraction.

If you divide $142 \, 857$ into two equal parts and add them, you get $999$:
 * $142 + 857 = 999$

Thus by Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999, $142 \, 857$ is divisible by $999$:
 * $142 \, 857 = 143 \times 999$

Also, we have:
 * $999 \, 999 = 1001 \times 999$

and so $999 \, 999$ is divisible by $999$.

But as $999 \, 999 = 7 \times 142 \, 857$ we have that $999 \, 999$ is divisible by $7$.

Thus it follows from Euclid's Lemma that $142 \, 857$ is divisible by $999$.