# Questions tagged [integers]

For questions about the set of integers, or positive and negative whole numbers, commonly denoted $\mathbb{Z}$.

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### If $3\mid mn$, then $3\mid m$ or $3\mid n$

I'm currently studying proofs and fundamentals, I'm reading a book by my own and I saw this problem. Theorem Let $m$ and $n$ be integers. If $3\mid mn$, then $3\mid m$ or $3\mid n$. My proof was the ...
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### Finding general term of sequence satisfying $f(m+n)+f(m-n)=\frac 12(f(2n) + f(2m))$ and $f(1) = 1$

The problem asks to compute $f(2020)$ knowing that $f(1) = 1$ and $f(m+n)+f(m-n)=\dfrac 12(f(2n) + f(2m))$ for integers $m,n$ such that $m>n\ge 1$. My try : I conjectured $f(n) = kn^2$ where $k$...
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### Solving a system of equations involving the floor function.

I have the following system of equations that I am stuggeling with: $$ax\lfloor y\rfloor=k,by\lfloor x\rfloor=d$$ And I know that $x$ and $y$ are bigger than zero and all the other constants are ...
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### Bouncing Bullet Problem

This is a problem that was presented to me through the google foobar challenge, but my time has since expired and they had decided that I did not complete the problem. I suspect a possible bug on ...
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### Is there a geometric proof for distributivity of integer addition/multiplication and other similar properties?

I am aware that commutativity, associativity and distributivity of integer addition and multiplication follow from their standard set theoretic definitions but I am looking for something suitable for ...
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### How do determine set of exclusions of sum given minimums, maximimums and sets of exclusions of the summed integers?

I'm trying to build a language compiler, but have come across a math-specific problem when implementing addition between integral types. I have n integers ...
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### Formula or algorithm needed related to integers sequences [closed]

Given a number "X" in such a spiral-like progression of integers in successive layers "n" what formula or algorithm is there to find the layer n in which it lies for very large X? (tip: you might ...
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### Given that $n^4-4n^3+14n^2-20n+10$ is a perfect square, find all integers n that satisfy the condition

So, I tried solving that by $$n^4-4n^3+14n^2-20n+10=x^2\\10=x^2-a^2, a^2=n^4-4n^3+14n^2-20n+10\\10=(x+a)(x-a)$$ but I couldn't find any integers when I solved it
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### find integer solutions under square root [duplicate]

I have a equation $y = \sqrt{5x^2+2x+1}$ and I'm trying to generate integer solutions. I've tried Vieta jumping but it failed. So I generated by brute force few solutions and find these: x=2, 15, ...
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### Analysis- Supremum and infimum

I tried to do this by taking $X=\{1,3,5,7\}$ and $Y =$ set of all odd natural numbers. In this case the inf$(A)$ is negative infinity. And sup$(A)$ is finite. But is it enough to answer the ...
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### If $xy=ab$ and $x<a\leq b<y$ then $x+y>a+b$.

Let $a,b,x,y$ be positive integers $>0$. Suppose \begin{align} xy&=ab,\\ x<a&\leq b<y \end{align}. Then how to show that $x+y>a+b$? I saw this statement in a comment in the ...
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### Optimize integer division

I have two positive integers $x$ and $y$. I need to calculate $\frac{100x}{x+y}$ and $\frac{100y}{x+y}$, which sum up to $100$ of course. However, I can only perform integer division. And since the ...
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### Regular polygon with integral ratio between apothem and side

Let $n$ be an integer. Find at least one $n$ such that the ratio between tha apothem and the side of a regular polygon with $n$ sides is an integer. I found this problem while I was casually playing ...
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### Variations of random coprime integers probability

The probability for two random integers to be coprime is $\frac{6}{\pi^2}$ (see for example this post), that is about $61\%$. After some computations, for $u_i, v_i$ random integers, the probability ...
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### Irreducibility of polynomials degree $3$

Let $f(x) = 2x^3+ax^2+bx+c$ where $a,b,c \in \Bbb Z$. Prove that $f$ is irreducible in $\Bbb Q[x]$ if and only if $f(d/2)$ does not equal $0$ for all $d \in \Bbb Z$. I'm not really sure how to start ...
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### Bijection of the set of natural numbers onto the set of integers. [closed]

An example in Real Analysis by Sherbert and Bartle tells that the set of integers is a bijection of the set of natural numbers. How is the one to one correspondence possible for the set of integers? ...
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### Find integer values that when multiplied together equal a given value

Given a = bc, with a known integer a, is it possible to find all b and c values that are integers quickly without testing each b and c value? As an example a = 194920496263521028482429080527, is it ...
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### Identifying a number is an integer, fraction or decimal.

Q- Is $\dfrac{24}{6}$ an integer,fraction,decimal? I think it is an integer and a fraction. Its an integer because $\dfrac{24}{6}$ can be reduced to get an integer. It is not decimal because it doesn'...
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### Relation between $x^{x+1}$ and $(x+1)^{x}$, $x \in \mathbb{Z}$

So say that we have a pair $(x^{x+1},(x+1)^{x})$ for all $x \in \mathbb{Z}$. Is there any correlation between the members of this pair? Or are they not related?
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### Is 5.0 an integer or decimal number?

Is 5.0 an integer or decimal number? I was asked by one of my friends, we got both confused. I said by definition integer contains no or zero decimal part so it should be an integer. But he said that ...
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### Proof explaination - $\sum_{i=1}^{n} \frac{1}{i}$ is not an integer for $n>1$

I was reading a proof to the following fact: for $n>1$, $\sum_{i=1}^{n} \frac{1}{i} \notin \mathbb{Z}$. The proof is as follows: Denote for prime $p$ by $v_p(a)$ the p-adic valuation of $a$. Write ...