# Tagged Questions

Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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### Check if sum is possible

Given a range $[L,R]$ I need to find weather a sum $S$ can be made by taking any number between this range i.e $L, L+1, L+2,\dotsc, R$ any number of times EXAMPLE: If $S=5$ and $L=2$ and $R=3$ then ...
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### Prove or disprove that if $a = bm + r$, then $\gcd(a,m) = \gcd(a,r)$

Prove or disprove that if $a = bm + r$, then $\gcd(a,m) = \gcd(a,r)$ I tried using the fact of GCD's in my calculations to get the fact that d|b and d|a-bm then try to compare that with the gcd(a,m) ...
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### if $x^2+ax+b=0$ has an integer root, show that it divides b [duplicate]

I don't know where to start. can anyone help me please ? if $x^2+ax+b=0$ has an integer root, show that it divides b
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### Complete Residue System Proof - n elements

Theorem: Let $n$ be a natural number. Every $complete$ $residue$ $system$ $modulo$ $n$ contains $n$ elements. The definition of a $complete$ $residue$ $system$ $modulo$ $n$ as given in our text: Let ...
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### Is there a commonly studied number structure with “two” of a given number?

I'm wondering whether there are structures of numbers where there is intuitively "two" of a given number. I have in mind something like what is illustrated in the following example number line ...
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### Show that in a ring R, if a*b=b*a=1 and a*c=1, then b=c

I have begun by showing that R is a commutative ring since the given shows that there exists an inverse of b so that a*b=1 and also that multiplication is commutative. Next I have sown that a is a ...
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### In a ring with identity, prove that (-1)(-1) = 1

I would like to attempt to solve this problem on my own if someone could give me a suggestion for how to start it.
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### Consider the equation $b^n=a^b$, where $a,b,n\in \mathbb{N}^{\ast}$

DISCLAIMER: This is a self-made problem, and I don't know if there's a nice solution. Hi, here's a conjecture I have for you; I'm not seeing the "simple" solution (at least for part (a)) at the ...
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### Problem with Diophantine equation

Let $a,b \in \mathbb N$ be coprime. Prove that for all $n\in \mathbb N$ such that $n>ab$ there are $r,s\in \mathbb N$ such that $n=ra+sb$. I'm really stuck on this problem. I know that since ...
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### Find all triples $(p; q; r)$ of primes such that $pq = r+ 1$ and $2(p ^ 2+q ^ 2) =r ^ 2 + 1$.

We have to find all triples $(p; q; r)$ of primes such that $pq = r+ 1$ and $2(p ^ 2+q ^ 2) =r ^ 2 + 1$. This question was asked in the 2013 mumbai region RMO but i could not find a solution to it. ...
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### Roots of functions / polynomials

Please excuse the naivity of this question, but it is a concept that I just have not been able to grasp entirely. My question is, why are the roots of a function, or a system of polynomials so ...
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### Position of 20096 in triangular array of all natural numbers

Write the set of all natural numbers in a triangular array as Find the row number and column number where $20096$ occurs. For example, $8$ occurs on row: $3$, column: $2$ Now, the upper row is ...
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### If distinct numbers $a,b,c\in\mathbb N^+$ satisfy $(a+b)(a+c)=(b+c)^2$, prove that $(b-c)^2>8(b+c)$.

If distinct numbers $a,b,c\in\mathbb N^+$ satisfy $$(a+b)(a+c)=(b+c)^2$$prove that $$(b-c)^2>8(b+c).$$ The first thing I did after I saw the problem was turning the inequality into this: ...
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### Find all $n\in\mathbb N^+$ such that the sum of the digits of $5^n$ is equal to $2^n$.

Find all $n\in\mathbb N^+$ such that the sum of the digits of $5^n$ is equal to $2^n$. I've solved this, but I think the proof is a bit weird and I wonder if there's a better one. My proof: ...
$\gcd(a,b)=1$ if and only if there is no prime $p$ such that $p|a$ and $p|b$ Prove it. So I went about doing it through contradiction: If $p|a$ and $p|b$ then $p|(x_{1})(x_{2})(x_{3})...$ where ...
$x_1, x_2,\ldots, x_{96}$ are positive integers greater than 2, which satisfy the relation: $$\frac{1}{x_1^4}+\frac{1}{x_2^4}+\cdots+\frac{1}{x_{96}^4}=\frac{1}{6}$$ I have two questions: 1. At least ...