When is This Fraction an Integer?

[richardfredlund]

oh i guess it must be if and only if, on the condition of (n^3-n )/(2^n+n) and the original question, so is final answer and doesn't need checking.

[mcwulf]

Very good. I solved it in much the same way.

[matthewfeig]

It would be very easy to extend this to a solution for all integers, not just positive integers. n=0 is a solution. For n <= -1, it's not hard to show the ratio is strictly between 1 and 2 (based off 0 < 8^n < 2^n < 1), so it's never an integer.

[willbishop]

When n=8 it should be 504/264, but either way not an integer.

[iainfulton]

The fraction becomes less than 2 at n = 8.

[cameronspalding]

When n=8, the numerator should be 264 not 262

[sy]

I miss to write conditions for one of the problems in the last comment. Regarding "Q4-(10)",
a+b+c+d+e = a^3+b^3+c^3+d^3+e^3 = 5 (a, b, c, d, e: integers) (a≧b≧c≧d≧e)
Find all (a, b, c, d, e). Find more than 6 solutions. (If there are less than 6 solutions, prove it.)
is the correct problem. The condition (a≧b≧c≧d≧e) was omitted. [ Excuse me. ]

[ΠασχάληςΜπαμπουλης]

Can you solve: x^n + (2+x)^n + (2-x)^n = 0, x, n integers

[mauriziosant]

The answer is then the aliens
greets.😎

[FadkinsDiet]

Why is m=0 ruled out? It looks like (1+0)/(1+0)=1

[infinity]

hi someone can help
me with this . just can’t seem to find the math method though i was able to brute force answer as 1232

Find the number of 2-element subsets {a, b} of {1, 2, 3, ..., 99, 100} such that ab + a + b is a multiple of 7.

[cyocs]

i dont get it why n >= 10, because [(4+1)/4]^3 is less then 2 or did i pluged it in the wrong equation. My english is bad so it can be that i didnt understand it correctly.

[sy]

Thank you for explaining. Today's problem is an interesting integer problem. If you want to treat integer problems, I would like to inform as below:

[information]
I am also creating and uploading mathematics videos like you, so I would like to introduce you. 　
In my case, with regard to mathematics, I had a time to study math for about only 1 to 2 years at university.
When I entered university, I passed the faculty (that students can study normal math) of a national university,
but my parents did not admit me to go to the university.
It is a bitter experience. Therefore, after reaching retirement age, I started creating and uploading mathematics videos.

However, I realized that there were some mistakes about using math words when I explain. (>_<)
Some of the corrections were not made in time, so they are uploaded as they are.
(Please accept this situation.)

Nowadays, I am planning as follows:
"I will publish only my own problems at first, and after various mathematics YouTubers introducing the methods in their videos,
I will upload a video of the answers that I have prepared. So, mathematics YouTubers can compete each other for how to solve.”

That's why I wrote this in this comment.
There are 40 questions in total (excluding the Appendix).
I prepared English versions and French versions for them. (But I am a Japanese. So Japanese explanation is used in my videos.)
For these problems, please refer to the following video.
[Video about 40 problems (Daily motion)]
SY_Math-Science_045 ( [Extra edition] The Special Event - 3 Second half) - Videos Dailymotion
[All my past videos (the 44th and 55th of these are related to the math problems)]
[Daily motion] → SY math science videos - Dailymotion

There are many number problems in the range of high school 10th (and 11th) grade and junior high school.
The problems were created based on the concept of "There are more than one solutions!"
Some of them are similar to some university entrance exams, and some of them are mathematical puzzles.
In addition, there are problems that can be solved by "the Try and Error method."
As for some problems, calculations are troublesome, concerning them, using calculator is admitted.
I guess you will be interested in some of them (or just a few?) if you like math.
If you find my problems interesting, please include them on your videos.
Instead, if you treat the problems that I made, please introduce the source of the problem and advertise my videos because I want to spread them.
(I need the propaganda.)

By the way, regarding video creation, it corresponds to question selection/creation, recording, BGM selection/input, and undercard puppet show,
editing (various cuts, text input, etc.), thumbnail creation/implementation, video management, etc. I do everything by myself.
Therefore, it takes more than a week from start creating to uploading per one video.
Therefore, I cannot make and upload my videos about something with related to them soon. (It takes time.)

"I apologize for writing so much information." Thank you very much for reading my comment.
(I look forward to your future success.) (^o^)

[Appendix]
Of the 40 problems (that I mentioned above), I introduce only 7 problems as samples below.
(Calculators are not admitted to use.) [The last one is not an integer problem.]

Q2-(2)
p^4 - 20p^3 + 90p^2 + 20p - 86 = q (p, q: prime number)　　
Find (p, q)

Q3-(8)
m^4 + 4m^2 + 8m + 9 = p^2 (m: integer, p: prime number)　
Find (m, p)

Q4-(3)
p^8 - 16q^8 = rst (p, q, r, s, t: prime number)
Find (p, q, r, s, t)

Q4-(5)
2m^3 + 4m^2n + 5m^2 + 4mn + 4n^2 + 2m + 4n = 2024 (m, n: integer)
Find (m, n)

Q4-(10)
a+b+c+d+e = a^3+b^3+c^3+d^3+e^3 = 5 (a, b, c, d, e: integer)
Find more than 6 solutions.
(If there are 6 or less than 6 solutions, prove it.)

Q5-(4)
2^a + 3^b + 4^c + 6^d = n! (a, b, c, d: integer, n: natural number)　
Find (a, b, c, d, n)

Q6-(2)
If x^12 + x^10 + 2x^8 + 2x^6 + x^4 - 8x^2 - 9 = 0,
what's x^10 + 2x^6 - 4x^5 + x^2 - 4x -77 ?
(x: complex number)

[nassirali]

I did another method and i found that (8^n+1)/(2^n+1)=4^n-2^n+1
And we know that 4^n-2^n+1 is an integer when n is a positive
Thus (8^n+1)/(2^n+1) is an integer with all n such that n is a positive integer