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

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59
votes
12answers
13k views

Dividing 100% by 3 without any left

In mathematics, as far as I know, you can't divide 100% by 3 without having 0,1...% left. Imagine an apple which was cloned two times, so the other 2 are completely equal in 'quantity'. The totality ...
33
votes
4answers
1k views

How does the divisibility graphs work?

I came across this graphic method for checking divisibility by $7$. $\hskip1.5in$ Write down a number $n$. Start at the small white node at the bottom of the graph. For each digit $d$ in ...
21
votes
10answers
1k views

Prove if $56x = 65y$ then $x + y$ is divisible by $11$

If $x$ and $y$ are natural numbers, and $56x = 65y$, prove that $x + y$ is divisible by $11$. I tried taking the $\gcd(56x,65y)$ using the Euclidean algorithm, but I got nowhere with it and do not ...
21
votes
5answers
9k views

If $a^2$ divides $b^2$, then $a$ divides $b$

Let $a$ and $b$ be positive integers. Prove that: If $a^2$ divides $b^2$, then $a$ divides $b$. Context: the lecturer wrote this up in my notes without proving it, but I can't seem to figure out ...
20
votes
1answer
475 views

Is $\sum_{k=1}^{n} k^k / \sum_{k=1}^{n} k \in \mathbb{N}$ for some $n > 1$?

Let $ A = \sum_{k=1}^{n} k^k $ and $ B = \sum_{k=1}^{n} k$, where $n >1 $ is a positive integer. Is $A/B$ ever an integer?
19
votes
5answers
3k views

Why $a^n - b^n$ is divisible by $a-b$?

I did some mathematical induction problems on divisibility $9^n$ $-$ $2^n$ is divisible by 7. $4^n$ $-$ $1$ is divisible by 3. $9^n$ $-$ $4^n$ is divisible by 5. Can these be generalized as $a^n$ ...
18
votes
4answers
1k views

Is the number $333, 333, 333, 333, 333, 333, 333, 333, 334$ a perfect square?

I know that if the number is a perfect square then it will be congruent to 0 or 1 (mod 4). Now since the number is even, I know that it is either 0 or 2 (mod 4). How would I go about answering this? ...
18
votes
5answers
1k views

How does one show that two general numbers $n! + 1$ and $(n+1)! + 1$ are relatively prime?

How does one show that two general numbers $n! + 1$ and $(n+1)! + 1$ are relatively prime? I don't mind if someone uses a different example, I want to learn how to prove this class of problems. My ...
18
votes
3answers
729 views

My first induction proof (very simple number theory). Please mark/grade.

What do you think about my first induction proof? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof First, introducing a predicate ...
17
votes
4answers
371 views

Prove that $\dfrac{(n^2)!}{(n!)^n}$ is an integer for every $n \in \mathbb{N}$

Prove that $\dfrac{(n^2)!}{(n!)^n}$ is an integer for every $n \in \mathbb{N}$ I know that there are tools in Number theory to proves the required but I want to use the tool that says that if you ...
14
votes
7answers
656 views

$n^5-n$ is divisible by $10$?

I was trying to prove this, and I realized that this is essentially a statement that $n^5$ has the same last digit as $n$, and to prove this it is sufficient to calculate $n^5$ for $0-9$ and see that ...
14
votes
6answers
1k views

Divisibility criteria of 24. Why is this?

I am currently familiar with the method of checking if a number is divisible by $2, 3, 4, 5, 6, 8, 9, 10, 11$. While Checking for divisibility for $24$ (online). I found out that the number has to ...
14
votes
2answers
748 views

Proving there are an infinite number of pairs of positive integers $(m,n)$ such that $\frac{m+1}{n}+\frac{n+1}{m}$ is a positive integer

The question is: Show that there are an infinite number of pairs $(m,n): m, n \in \mathbb{Z}^{+}$, such that: $$\frac{m+1}{n}+\frac{n+1}{m} \in \mathbb{Z}^{+}$$ I started off approaching this ...
14
votes
3answers
426 views

Prove $n\mid \phi(2^n-1)$

If $2^p-1$ is a prime,(thus $p$ is a prime,too) then $p\mid 2^p-2=\phi(2^p-1).$ But I find $n\mid \phi(2^n-1)$ is always hold, no matter what $n$ is.Such as $4\mid \phi(2^4-1)=8.$ If we denote ...
13
votes
5answers
3k views

Proof of the divisibility rule of 17.

Rule: Subtract 5 times the last digit from the rest of the number, if the result is divisible by 17 then the number is also divisible by 17. How does this rule work? Please give the proof. ...
13
votes
2answers
280 views

Prove $6 \nmid [\left( \sqrt[3]{28} - 3 \right)^{-n}]$

Prove that: $$6 \not\left|\ \left\lfloor\frac 1 {(\sqrt[3]{28} - 3)^{n}}\right\rfloor \ (n \in Z^+)\right.$$ ($\lfloor x\rfloor$ = largest integer not exceeding $x$) I am very bad as English and ...
13
votes
1answer
452 views

My attempt to prove GCD exists

Please review my attempt to prove a theorem. Any mistakes you point would be highly appreciated by me. To prove the theorem, I'll be using the following properties which I'm assuming have already ...
12
votes
7answers
448 views

How can I find the possible values that $\gcd(a+b,a^2+b^2)$ can take, if $\gcd(a,b)=1$

If $\gcd(a,b)=1$, how can I find the values that $\gcd(a+b,a^2+b^2)$ can possibly take? I can´t find a way to use any of the elemental divisibility and gcd theorems to find them.
12
votes
1answer
368 views

Do there exist two primes $p<q$ such that $p^n-1\mid q^n-1$ for infinitely many $n$?

We can prove that there is no integer $n>1$ such that $2^n-1\mid 3^n-1$. This leads to the following question: Is it true that for every pair of primes $p<q$ there are only finitely many ...
11
votes
2answers
331 views

How rare are the primes $p$ such that $p$ divides the sum of all primes less than $p$?

This is just for fun! The title pretty much says it all. It's probably a very difficult question. Up to the $40,000^{th}$ prime $(479909)$, I have found only $5$, $71$ and $369119$ with this ...
11
votes
1answer
184 views

Family of GCDs all equal to $2$

Is it true that $$\gcd\left(5^{2^n} + 1, 13^{2^n} + 1\right) = 2$$ for all $n \in \mathbf{Z}_{\geq 0}$? I'm continually stumped with this and verifying it numerically is quite expensive very ...
11
votes
2answers
598 views

Does $n \mid 2^{2^n+1}+1$ imply $n \mid 2^{2^{2^n+1}+1}+1$?

There are two ways to try to prove this. One is in the title, the other is its de Morgan counterpart: $n \nmid 2^{2^{2^n+1}+1}+1 \implies n \nmid 2^{2^n+1}+1$. Disproving it requires only one example ...
11
votes
1answer
217 views

Curious number theory problem

$k,m,n\in\mathbb{N}$ satisfy $k^{m+n}=nm^n$. How can I show that $m=k$ and $n=k^k\,?$
10
votes
5answers
2k views

Disproving the claim that the numbers 1+2+4, 1+2+4+8, 1+2+4+8+16… alternate between prime and composite

I am working through an elementary number theory book and I have come across the following problem. Show the following claims are wrong: Claim 1: The sequence 1+2+4, 1+2+4+8, 1+2+4+8+16, ...
10
votes
4answers
436 views

Middle school number theory

Find at least three numbers that satisfy all three conditions: (1) there is a remainder of $1$ when the number is divided by $2$; (2) there is a remainder of $2$ when the number is divided by $3$; ...
10
votes
4answers
293 views

Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$

I can't crack this one. Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ (the only constraint is that $a,b,c,z \in \mathbb{Z}$).
10
votes
1answer
116 views

Find all $x,y,z\in\mathbb N$, $x,y,z>1$ such that satisfy $x\mid yz+1$, $y\mid xz+1$, and $z\mid xy+1$

Find all $x,y,z\in\mathbb N$, $x,y,z>1$ such that satisfy $$\begin{cases}x\mid yz+1\\y\mid xz+1\\z\mid xy+1\end{cases}$$ I've found out easily that $$\begin{cases}x\nmid yz\\y\nmid xz\\z\nmid ...
10
votes
2answers
185 views

If $n\in\mathbb N$ and $4^n+2^n+1$ is prime, prove that there exists an $m\in\mathbb N\cup\{0\}$ such that $n=3^m$.

If $n\in\mathbb N$ and $4^n+2^n+1$ is prime, prove that there exists an $m\in\mathbb N\cup\{0\}$ such that $n=3^m$. I.e. if $4^n+2^n+1$ is prime, prove that $n=3^m$, where $m\in\mathbb N\cup\{0\}$. ...
9
votes
6answers
2k views

Show that $11^{n+1}+12^{2n-1}$ is divisible by $133$.

Problem taken from a paper on mathematical induction by Gerardo Con Diaz. Although it doesn't look like anything special, I have spent a considerable amount of time trying to crack this, with no luck. ...
9
votes
6answers
250 views

Understanding the proof of a formula for $p^e\Vert n!$

This is a proof from a book on number theory I'm reading. I'm having a hard time following. I think there's a variable here that means two different things at two different times... Theorem: If n is ...
9
votes
3answers
231 views

Show that $(n!)^{(n-1)!}$ divides $(n!)!$

Show that $(n!)^{(n-1)!}$ divides $(n!)!$ I found this question in a text he was reading about BREAKDOWN OF PRIME FACTORS IN FACT, I decided many years, have posted some here to help, but this ...
9
votes
3answers
284 views

Does $a^n \mid b^n$ imply $a\mid b$?

Does $a^n \mid b^n$ imply $a\mid b$? I think it does but haven't been able to prove it. I don't know much number theory so an elementary answer would be great.
9
votes
1answer
95 views

For what integers $n$ is this divisibility statement true?

The statement being $$n^2 + 2 \mid 2014n + 2$$ The answer is $n = -2, 0, 1, 2014$. Don't know how to arrive at this answer without using comp sci. (Using the compsci answer, we can restrict the ...
9
votes
1answer
110 views

A divisibility question concerning positive integers

Suppose $n$ is a positive integer such that $3n+1$ and $4n+1$ are both perfect squares , then how do we prove that $7|n$ ?
9
votes
1answer
185 views

Elementary Number Theory; prove existence

Prove that there exists a positive integer $n$ such that $$2^{2012}\;|\;n^n+2011.$$ I was wondering if you could prove this somehow with induction (assume that $n$ exists for $2^k|n^n+2011$ ...
8
votes
3answers
691 views

Prove that $x$ and $x+1$ are coprime numbers

Given $\{x \mid x > 1\}$, how do I prove that any given $x$ and $x+1$ are coprime?
8
votes
7answers
821 views

Better Divisibility by 8

Everywhere I look, when you want to see if something is divisible by $8$ then you see if the last $3$ digits are divisible by eight. But how do you know if the last $3$ digits are divisible by $8$? ...
8
votes
7answers
1k views

Proof for divisibility by $7$

One very classic story about divisibility is something like this. A number is divisible by $2^n$ if the last $n$-digit of the number is divisible by $2^n$. A number is divisible by 3 (resp., by ...
8
votes
5answers
303 views

Prove that $b\mid a \implies (n^b-1)\mid (n^a-1)$

Given natural numbers $a,b,n$, prove $b\mid a \implies (n^b-1)\mid (n^a-1)$. I tried the simple method of beginning with $b\mid a \implies \exists k \in \mathbb{N} $ such that $bk=a$ and then ...
8
votes
4answers
439 views

Why does $ (\frac{1}{2})^∞ = 0?$

Recently while at my tutoring I had a question that said: "Aladin has a pair of magic scissors that can cut things in to tiny pieces. If he cuts a carpet in half, cuts the half into half and continues ...
8
votes
3answers
358 views

Proof of Wolstenholme's theorem.?

According to the theorem : $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...+\frac{1}{p-1} =\frac{r}{q}$$ And we have to prove that $r= 0 \pmod{p^2}$. (Given $ p>3$, ...
8
votes
4answers
795 views

Remainder when $20^{15} + 16^{18}$ is divided by 17

What is the reminder, when $20^{15} + 16^{18}$ is divided by 17. I'm asking the similar question because I have little confusions in MOD. If you use mod then please elaborate that for beginner. ...
8
votes
2answers
2k views

Divisibility Rules for Bases other than $10$

I still remember the feeling, when I learned that a number is divisible by $3$, if the digit sum is divisible by $3$. The general way to get these rules for the regular decimal system is ...
8
votes
1answer
626 views

If $\gcd(a,b)=d$, then $\gcd(ac,bc)=cd$?

$A$ an integral domain, $a,b,c\in A$. If $d$ is a greatest common divisor of $a$ and $b$, is it true that $cd$ is a greatest common divisor of $ca$ and $cb$? I know it is true if $A$ is a UFD, but ...
8
votes
3answers
160 views

How can we find the gcd for elements (binomial coefficient)?

$\gcd\left(\binom{2n}1,\binom{2n}3,\binom{2n}5,\ldots,\binom{2n}{2n-1}\right)$ i want to know what is specialty of such a series.I am not able to generalize the problem solution.Is there any rule for ...
8
votes
3answers
107 views

g.c.d.{$m^p-m:m=2$ to $n$} $= ?$

Let $p$ be an odd prime and $n>2$ is an integer , then what is the $g.c.d.$ of the numbers {$m^p-m:m=2$ to $n$} ? (by Fermat's little theorem it is easy to see , $2p$ divides the g.c.d. , but I can ...
8
votes
1answer
423 views

$f'/f\in\mathbb{Z}[[x]]$ for polynomials vs. formal power series $f$

I am curious about the following problem from MIT's Problem Solving Seminar (#26 here, though the link may stop working after a few weeks): Let $f(x) = a_0+a_1x+\cdots\in\mathbb{Z}[[x]]$ be a ...
8
votes
1answer
107 views

GCD of $a^n + b^n$ and $c^n + d^n$

Prove or disprove that there does not exists any integers $a,b,c,d > 1$ such that $a,b,c,d$ are pairwise coprime, and $a^n + b^n$ and $c^n + d^n$ are also coprime for all integer $n > 1$. I ...
7
votes
11answers
371 views

3 never divides $n^2+1$

Problem: Is it true that 3 never divides $n^2+1$ for every positive integer n? Explain. Explanation: If n is odd, then $n^2+1$ is even. Hence 3 never divides $n^2+1$, when n is odd. If n is even, ...
7
votes
8answers
268 views

Why is $2x^3 + x$, where $x \in \mathbb{N}$, always divisible by 3?

So, do you guys have any ideas? Sorry if this might seem like dumb question, but I have asked everyone I know and we haven't got a clue.