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NT marathon!!!!!!!
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I am noob in NT. if the answer is wrong please share me the right solution
- Mehrab4226
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Re: Problem 7
Anindya Biswas wrote: ↑Wed Mar 24, 2021 3:26 pmLet $a,b, c, d$ be integers. Show that the product \[(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)\] is divisible by $12$
Source :
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré
-Henri Poincaré
- Mehrab4226
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Problem: 8
Prove that any integer greater than or equal to $7$ can be written as a sum of two relatively prime integers, both greater than $1$.
Source:
Source:
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré
-Henri Poincaré
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Re: Problem: 8
wow this was a brain teaser (sorry saw from mathstacks cuz i am noob )Mehrab4226 wrote: ↑Wed Mar 24, 2021 7:44 pmProve that any integer greater than or equal to $7$ can be written as a sum of two relatively prime integers, both greater than $1$.
Source:
if $n=2k+1$ then $k$ and $k+1$ are coprime.
if $n=4k$ then $2k-1$ and $2k+1$ are coprime
if $n=4k+2$ then $2k-1$ and $2k+3$ are coprime. $\square$
Hmm..Hammer...Treat everything as nail
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Problem 9
An old question nobody answered that so reposted
Problem 9
Find all $a,b \in \mathbb{N}$ such that $y=ax^2+6x+b$ and $y=ax+6$ intersect only once.
Problem 9
Find all $a,b \in \mathbb{N}$ such that $y=ax^2+6x+b$ and $y=ax+6$ intersect only once.
Hmm..Hammer...Treat everything as nail
- Mehrab4226
- Posts:230
- Joined:Sat Jan 11, 2020 1:38 pm
- Location:Dhaka, Bangladesh
Re: Problem 9
There are too many calculations(so a lot of chances for mistakes by me. So I might have skipped some parts. If it is difficult to understand, feel free to ask.Asif Hossain wrote: ↑Fri Mar 26, 2021 7:54 amAn old question nobody answered that so reposted
Problem 9
Find all $a,b \in \mathbb{N}$ such that $y=ax^2+6x+b$ and $y=ax+6$ intersect only once.
Last edited by Mehrab4226 on Fri Mar 26, 2021 2:54 pm, edited 1 time in total.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré
-Henri Poincaré
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- Posts:194
- Joined:Sat Jan 02, 2021 9:28 pm
Re: Problem 9
Recheck your proof There are solutions for a,b one example is $(a,b)=(2,8)$Mehrab4226 wrote: ↑Fri Mar 26, 2021 2:23 pmThere are too many calculations(so a lot of chances for mistakes by me. So I might have skipped some parts. If it is difficult to understand, feel free to ask.Asif Hossain wrote: ↑Fri Mar 26, 2021 7:54 amAn old question nobody answered that so reposted
Problem 9
Find all $a,b \in \mathbb{N}$ such that $y=ax^2+6x+b$ and $y=ax+6$ intersect only once.
Hmm..Hammer...Treat everything as nail
- Mehrab4226
- Posts:230
- Joined:Sat Jan 11, 2020 1:38 pm
- Location:Dhaka, Bangladesh
Re: Problem 9
Thank you for pointing that out. I used so the quadratic formula so many times that I accidentally forgot to give the minus before b in a line. I updated my solution.Asif Hossain wrote: ↑Fri Mar 26, 2021 2:39 pmRecheck your proof There are solutions for a,b one example is $(a,b)=(2,8)$Mehrab4226 wrote: ↑Fri Mar 26, 2021 2:23 pmThere are too many calculations(so a lot of chances for mistakes by me. So I might have skipped some parts. If it is difficult to understand, feel free to ask.Asif Hossain wrote: ↑Fri Mar 26, 2021 7:54 amAn old question nobody answered that so reposted
Problem 9
Find all $a,b \in \mathbb{N}$ such that $y=ax^2+6x+b$ and $y=ax+6$ intersect only once.
Here is a meme as a token of appreciation,
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré
-Henri Poincaré
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- Posts:194
- Joined:Sat Jan 02, 2021 9:28 pm
Re: Problem 10
Problem 10Asif Hossain wrote: ↑Fri Mar 26, 2021 7:54 amAn old question nobody answered that so reposted
Problem 9
Find all $a,b \in \mathbb{N}$ such that $y=ax^2+6x+b$ and $y=ax+6$ intersect only once.
Extend this to $\mathbb{Z}$
Hmm..Hammer...Treat everything as nail
- Mehrab4226
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- Location:Dhaka, Bangladesh
Solution of Problem 10
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré
-Henri Poincaré