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:

\(\ds 142 \, 857 \times 12\)

\(=\)

\(\ds 1 \, 714 \, 284\)

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 $7$:

\(\ds 142 \, 857 \times 7\)

\(=\)

\(\ds 999 \, 999\)

\(\ds 142 \, 857 \times 14\)

\(=\)

\(\ds 1 \, 999 \, 998\)

From Number times Recurring Part of Reciprocal 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 $999$.

This needs considerable tedious hard slog to complete it. In particular: A lot of material from this chapter of Wells has been skipped, because it's just not very interesting. I leave it open for someone else to complete, if they want to. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $142,857$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $142,857$