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# Questions tagged [divisibility]

This tag is for questions about divisibility, that is, determining when one thing is a multiple of another thing.

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### If prime power divides product of two consecutive integers then, it divides one of them? [duplicate]

Question: If $p^n$ divides product of two consecutive positive integers say, $m(m-1)$ then $\;$ $p^n\lvert\;{m}$ $\;$or $\;$ $p^n\;\lvert\;{m-1}$. For example $2^3\lvert\;m(m-1)$ then $2^3\;\lvert\;m$ ...
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1 answer
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### Does a infinite a.p with general modulus always contain a subsequence in which any 2 elements are coprime [closed]

Let $A=\{am_1+b, am_2+b, am_3+b,...\}$. for $a,b$ coprime integers, $a \gt 1$, and $m_i$ being a sequence of strictly increasing positive integers. Does $A$ always contain an infinite subsequence in ...
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### Divisibility of $a^m + b^m$ by $a^n + b^n$ implies divisibility of m by n [duplicate]

I'm working on a problem that states: Let $a, b, m, n$ be natural numbers, $a > 1$, and suppose that $a$ and $b$ have no common factors (GCD). We are asked to prove that if $a^m + b^m$ is divisible ...
1 vote
1 answer
154 views

### Understanding the motivation of a step in a proof

I need help understanding the proof of a divisibility problem I was trying, it is not that I do not understand the steps of the proof, but rather I'm having a hard time understanding the motivation of ...
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1 vote
0 answers
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### Confusion for algorithm for finding (a div d) and (a mod d), where a is an integer and d positive integer.

From Rosen's discrete Math textbook. I'm confused on 3 things regarding this algorithm (as can be seen via the screenshots) Why do we need an algorithm for finding $a$ div $d$ and $a$ mod $d$ when we ...
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3 votes
2 answers
96 views

### Determine the value of the sum of all elements of the set $A=\{ x \in \Bbb R | \frac{3x^2+5x+2}{x^2+x+1}\in \Bbb N\}$

The problem Determine the value of the sum of all elements of the set $A=\{ x\in \Bbb R : \frac{3x^2+5x+2}{x^2+x+1}\in \Bbb N\}$ my idea I wrote $3x^2+5x+2=(3x+2)(x+1)$ For $\frac{3x^2+5x+2}{x^2+x+1}$ ...
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1 answer
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• 760
1 vote
1 answer
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### Conjectures about the greatest common divisor of a vertical column of the pascal triangle.

I was playing around with pascal triangle I noticed an interesting property concerning the greatest common divisor $gcd$ of binomial coefficients along a vertical line. Specifically, the line ...
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0 votes
2 answers
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### Odd numbers and divisibility by $9999.....9999$ - an analysis. [duplicate]

I am currently interested in the problem For an integer $m$, define $f(m)$ as the smallest integer $n$ such that $m \ | \ \overbrace{9999\dots9999}^{n}$ This is a property I observed after messing ...
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4 votes
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### Is the greatest prime factor of $2^n+1$ greater than or equal to $n$. [duplicate]

I had this question because I had to prove for a problem that $2^n+1 \not \mid (n-1)!$, so I thought it would be nice for there to be a prime factor of $2^n+1$ that is greater than $n-1$. I checked ...
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### For a given $m$, which is relatively prime to $p$, show there exists an $r$ such that $m$ divides $p^r-1.$ [duplicate]

My question is: Suppose that we have an arbitrary natural number $m$ that is not divisible by the prime number $p.$ Then there exists an integer $r$ such that $m$ divides $p^r-1$. Some help on this ...
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3 votes
2 answers
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### Divisibility (algebraic expressions) - can this be generalised?

Consider the expression $a^{3}+b^{3}+c^{3}-3abc$. It is divisible by $(a+b+c)$ Now, consider the expression $a^{3}+b^{3}+c^{3}+d^{3}-3(abc+abd+acd+bcd)$ It is divisible by $(a+b+c+d)$ Can this be ...
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1 vote
1 answer
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### Casting out nines and digital sums for fractions (rationals) [closed]

I am a high school student and there is something I want to ask about the application of digital sums. Let's say there is a fraction "520/7", let 520/7=a, so 520= a × 7, so if we now ...
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### Bound of squarefree part of an integer

I am studying the paper DIOPHANTINE EQUATIONS OF THE FORM $F(X) = G(Y)$ - AN EXPOSITION which discusses the result of Erdos and Selfridge. I am unable to understand the highlighted statement ''Clearly ...
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6 votes
1 answer
386 views

### 37 and Veritasium

In Veritasium's new video about 37 there is brought up something interesting about its multiples. For any multiple of 37 reverse it and put a 0 between all of its digits and the new number will be a ...
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4 votes
0 answers
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### Minimum $k$ for which every positive integer of the interval $(kn, (k+1)n)$ is divisible by at least one prime number less than $n$

As a continuation of this question relating the Minimum $k$ for which every positive integer of the interval $(kn, (k+1)n)$ is composite and this other one on the divisibility of numbers in intervals ...
• 1,190
3 votes
1 answer
227 views

### Can this divisibility be proven in general?

Let $n$ be a positive integer with $n\equiv 4\mod 6$ and define $p:=\frac{n^2+n+1}{3}$ (which is in this case a positive integer as well). Conjecture : $$p^2\mid n^n+(n+1)^{n+1}$$ for every $n$ of ...
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