# What makes the numbers 142857 special?

What makes the numbers 142857 special?

1/7 = 0.14285714285714..

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arildno
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Well, 142+857=999, and I would assume that 142999857 is divisible with 142,999 and 857 (I haven't checked the divisibility criterion though)

142+857=999 is not part of the intended solution

arildno
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But the rest is?

Master of the Obvious

AntonVrba said:
What makes the numbers 142857 special?

1/7 = 0.14285714285714..
:uhh: It only repeats as long as numbers exist??? :surprised

:uhh: Individually they are all less than 10??? :surprised

:uhh: It is a repeating sequence??? :surprised

:uhh: 1+4-2+8-5+7 = 13 and 1-4+2-8+5-7 = -11 with 13-11 = 2 which is "14"/7 and "28"/7 = 4 which is twice 2 and "57" doesn't matter at all because it already has a "7" in it.

It is one of the only short sequence repeating numbers???

Rahmuss said:
:uhh: 1+4-2+8-5+7 = 13 and 1-4+2-8+5-7 = -11 with 13-11 = 2 which is "14"/7 and "28"/7 = 4 which is twice 2 and "57" doesn't matter at all because it already has a "7" in it.

It is one of the only short sequence repeating numbers???
nice try but none of these or others above

but I like the bit about repeating but what repeats?

James R
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For example:

1/17 = 0.0588235294117647 0588235294117647 0588235294117647 ...
2/17 = 0.1176470588235294 1176470588235294 1176470588235294 ...
3/17 = 0.1764705882352941 1764705882352941 1764705882352941 ...

Notice the cyclic nature of the decimal expansion.

AntonVrba said:
nice try but none of these or others above

but I like the bit about repeating but what repeats?
Well... the "142857" repeats; but that isn't really too special. I'm really either not sure what you're asking, or I'm stumped.

OK - her is the answer

2 x 142857 = 285714
3 x 142857 = 428571
upto
6 x 142857= 857124

the product 2x, 3x, 4x, 5x, 6x of 142857 is always a combination of the same numbers, irrespective of how many times you repeat the nine in the middle

Wow! That is kind of neat. Go into the number theory forum and see if they can give a good reason as to why that is. Or do you know why that is?