7/(n+9), 8/(n+10),9/(n+11),…. 31/(n+33)

[japiga]

Find the least natural number n, so that anyone bellow divisions may not be shorten:

7/(n+9), 8/(n+10),9/(n+11),…. 31/(n+33)

I found that such number n = 17, but how to prove it?

[stapel_eliz]

What is meant by "shortening" a division?

[japiga]

Yes, sorry. "shorten" means "divided not to be further divided". Or generally, not to be divided at all. Is it now more clear?

[stapel_eliz]

Yes, sorry. "shorten" means "divided not to be further divided". Or generally, not to be divided at all. Is it now more clear?

Sorry; no.

[Martingale]

Find the least natural number n, so that anyone bellow divisions may not be shorten:

7/(n+9), 8/(n+10),9/(n+11),…. 31/(n+33)

I found that such number n = 17, but how to prove it?

If I understand you then 17 doesn't work...

[Martingale]

Yes, sorry. "shorten" means "divided not to be further divided". Or generally, not to be divided at all. Is it now more clear?

Sorry; no.

I believe that japiga wants the smallest

such that





Though I could be wrong

[japiga]

Yes, I am wrong, sorry. Thanks Martingale you are correct, but What is solution? What is the least number for n, such as that each division is not to be further divided for all line: 7/(n+9), 8/(n+10),9/(n+11),…. 31/(n+33) ?
And, to add, solution for "n" should be any integer number, but we are looking for the least one ("the smallest one").

[Martingale]

Yes, I am wrong, sorry. Thanks Martingale you are correct, but What is solution? What is the least number for n, such as that each division is not to be further divided for all line: 7/(n+9), 8/(n+10),9/(n+11),…. 31/(n+33) ?
And, to add, solution for "n" should be any integer number, but we are looking for the least one ("the smallest one").

Note:

for

if n=k-2 we get



that should eliminate a lot of possibilities.

or similarly...

if k=n+2

[japiga]

Thanks! It make sense if final result is 1/2, but I am little confused with original assignment: 7/(n+9), 8/(n+10),9/(n+11),…. 31/(n+33). Why he have line only with those numbers, not for all, like 32/(n + 34), 33/(n+35).... It seems that this rule is valid for any k/(n+k+2). Do you think so?

[Martingale]

Thanks! It make sense if final result is 1/2, but I am little confused with original assignment: 7/(n+9), 8/(n+10),9/(n+11),…. 31/(n+33). Why he have line only with those numbers, not for all, like 32/(n + 34), 33/(n+35).... It seems that this rule is valid for any k/(n+k+2). Do you think so?

if you were allowed to use all the numbers (ie get stuff like 32/(n + 34), 33/(n+35)...) then there would be no value of n that worked (by what I showed), but here we are only allowed to go to 31/(n+33). This restricts things ...making it so there is a solution.