# Questions tagged [elementary-number-theory]

Questions on divisibility, gcd and lcm, congruences, linear Diophantine equations, Fermat's and Wilson's theorems, the Chinese Remainder theorem, primitive roots, quadratic congruences, quadratic number fields and other related topics which may be treated in first courses on number theory. More advanced topics should instead use the number-theory or other tags.

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### Finding (a,p) s.t p|(1+a+...+a^6)

Let A=1+a+a^2+...+a^6 , where a>1, p an odd prime and p|A. The goal is to show that p=7 or p=1(7) I first started by showing that a^7=1(p) : A=1+a^2+...+a^6=(1/7).(a^7-1)=0(p) => a^7=1(p) And ...
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### Prove that for every sufficiently large $n$, we can write the positive integer $n$ in the form $n = a_1+ a_2+...+a_{2018}$ with $: (a_i , a_j ) = 1$

Prove that for every sufficiently large $n$, we can write the positive integer $n$ in the form $n = a_1+ a_2+...+a_{2018}$ In there $: (a_i , a_j ) = 1$ for all $i<j$ and $a_i>2018$ for all $i$ ...
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### Is it possible to easily calculate the resulting number if I add a random integer to all of the prime factors of a given number?

If I have an composite integer called $x$, is it possible for me to calculate what the resulting number will be if I add a random integer, denoted by $n$, to all of the factors of $x$ (without ...
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### Show that if $n \in N$ and $a,b \in \mathbb{ Z}$ s.t. $a+b=n$, then $\overline{a}, \overline{b}$ are additive inverses in $\mathbb{ Z_n}$.

If $a+b=n$, then in modulo remainder class have $\overline{a} + \overline{b}= 0$. So, $\overline{a} = -\overline{b}$. Hence, proved. Is it this simple, or am missing something?
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### How to think a solution for the problem like this? [duplicate]

The question: If (a/b)<(c/d) with b>0, d>0 show that (a+c)/((b+d) lies between a/b and c/d where a,b,c,d are real numbers Answer to the problem: (a/b)<(c/d) which implies ac<bd. Add ...
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### Show that a natural number n> 1 has no more than log2 (n) prime divisors. [closed]

a) Show that a natural number n> 1 has no more than log2 (n) prime divisors. b)Show that the odd natural number n> 1 has no more than log3 (n) prime divisors
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### Let $m$ and $n$ be positive integers such that $m = 24n + 51$. What is the largest possible value of the greatest common divisor of $2m$ and $3n$? [duplicate]

Let $m$ and $n$ be positive integers such that $m = 24n + 51$. What is the largest possible value of the greatest common divisor of $2m$ and $3n$? I'm trying to figure out how to use the Euclidean ...
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### Fermat's factorization method [closed]

In the case of $p$ is a prime number and $q$ is another prime number If $pq=n$ And $n$ was known in that equation. Is Fermat's method the best known method now for finding $p$ and $q$, or is there a ...
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### Do three consecutive numbers of form $A^2B^3$ exist?

I (non-mathematician) asked a similar kind of question 5 days ago. Now I revisit the case in a different manner. The powerful numbers may be written in the form $A^2B^3$, where $A$ and $B$ are ...
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### What is the greatest possible value of the greatest common divisor of $4T_n$ and $n-1$? [duplicate]

For all positive integers $n$, the $n$th triangular number $T_n$ is defined as $T_n = 1+2+3+ \cdots + n$. What is the greatest possible value of the greatest common divisor of $4T_n$ and $n-1$? Since ...