NT marathon!!!!!!!
- Mehrab4226
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Problem:11
Show that for all positive integers $n$, $81$ divides $10^{n+1}-9n-10$
Show that for all positive integers $n$, $81$ divides $10^{n+1}-9n-10$
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
- Anindya Biswas
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Re: Problem:11
Mehrab4226 wrote: ↑Mon Mar 29, 2021 10:33 amProblem:11
Show that for all positive integers $n$, $81$ divides $10^{n+1}-9n-10$
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann
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Re: Problem 12
SInce no body posting probs that's why here is one
Find all $n,a \in \mathbb{N}$ Such That $n!+25=a^2$
Source: a brother of mine
Find all $n,a \in \mathbb{N}$ Such That $n!+25=a^2$
Source: a brother of mine
Hmm..Hammer...Treat everything as nail
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- Posts:194
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Re: Problem 12
Solution (maybe idk )Asif Hossain wrote: ↑Fri Apr 02, 2021 10:34 amSInce no body posting probs that's why here is one
Find all $n,a \in \mathbb{N}$ Such That $n!+25=a^2$
Source: a brother of mine
Hmm..Hammer...Treat everything as nail
- Mehrab4226
- Posts:230
- Joined:Sat Jan 11, 2020 1:38 pm
- Location:Dhaka, Bangladesh
Problem:13
For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe.
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
- Anindya Biswas
- Posts:264
- Joined:Fri Oct 02, 2020 8:51 pm
- Location:Magura, Bangladesh
- Contact:
Solution to problem:13
Mehrab4226 wrote: ↑Wed Apr 07, 2021 10:08 amFor a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe.
Source:
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann
- Anindya Biswas
- Posts:264
- Joined:Fri Oct 02, 2020 8:51 pm
- Location:Magura, Bangladesh
- Contact:
Problem 14
Let $a_1,a_2,a_3,\dots$ be a sequence of positive integers such that $\text{gcd}(a_m,a_n)=\text{gcd}(m,n)$ where $m\neq n$ and $m,n$ positive integers. Prove that $a_m=m$ for all positive integer $m$.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann
Re: Problem 14
$\textbf{Solution 14}$Anindya Biswas wrote: ↑Thu Apr 08, 2021 10:23 amLet $a_1,a_2,a_3,\dots$ be a sequence of positive integers such that $\text{gcd}(a_m,a_n)=\text{gcd}(m,n)$ where $m\neq n$ and $m,n$ positive integers. Prove that $a_m=m$ for all positive integer $m$.
Last edited by ~Aurn0b~ on Thu Apr 08, 2021 2:56 pm, edited 1 time in total.
Re: NT marathon!!!!!!!
$\textbf{Problem 15}$
Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that:
\[ f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q \] Holds for all $p,q\in\mathbb{P}$.
Let $\mathbb{P}$ be the set of all prime numbers. Find all functions $f:\mathbb{P}\rightarrow\mathbb{P}$ such that:
\[ f(p)^{f(q)}+q^p=f(q)^{f(p)}+p^q \] Holds for all $p,q\in\mathbb{P}$.
- Mehrab4226
- Posts:230
- Joined:Sat Jan 11, 2020 1:38 pm
- Location:Dhaka, Bangladesh
Re: NT marathon!!!!!!!
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