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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.

123
votes
7answers
23k views

Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$

For all $a, m, n \in \mathbb{Z}^+$, $$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
118
votes
8answers
19k views

Are half of all numbers odd?

Plato puts the following words in Socrates' mouth in the Phaedo dialogue: I mean, for instance, the number three, and there are many other examples. Take the case of three; do you not think it may ...
114
votes
21answers
25k views

How do you explain to a 5th grader why division by zero is meaningless?

I wish to explain my younger brother: he is interested and curious, but he cannot grasp the concepts of limits and integration just yet. What is the best mathematical way to justify not allowing ...
100
votes
9answers
8k views

Division by $0$ and its restrictions

Consider the following expression: $$\frac{1}{2} \div \frac{4}{x}$$ Over here, one would state the restriction as $x \neq 0 $, as that would result in division by $0$. But if we rearrange the ...
83
votes
4answers
3k views

How to solve these two simultaneous “divisibilities” : $n+1\mid m^2+1$ and $m+1\mid n^2+1$

Is it possible to find all integers $m>0$ and $n>0$ such that $n+1\mid m^2+1$ and $m+1\,|\,n^2+1$ ? I succeed to prove there is an infinite number of solutions, but I cannot progress anymore. ...
82
votes
6answers
9k views

Any rectangular shape on a calculator numpad when divided by 11 gives an integer. Why?

I have come across this fact a very long time ago, but I still can't tell why is this happening. Given the standard calculators numpad: 7 8 9 4 5 6 1 2 3 if you ...
79
votes
9answers
4k views

Divisibility by 7 rule, and Congruence Arithmetic Laws

I have seen other criteria for divisibility by 7. Criterion described below present in the book Handbook of Mathematics for IN Bronshtein (p. 323) is interesting, but could not prove it. Let $n = (...
73
votes
14answers
83k 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 'quality'. The totality ...
72
votes
16answers
38k views

For any prime $p > 3$, why is $p^2-1$ always divisible by 24?

I know this is very basic and old hat to many, but I love this question and I am interested in seeing whether there are any proofs beyond the two I already know.
66
votes
6answers
10k views

Why is $9$ special in testing divisiblity by $9$ by summing decimal digits? (casting out nines)

I don't know if this is a well know fact but I have observed that every number no matter how large that is equally divided by $9$, will equal $9$ if you add all the numbers it is made from until there ...
64
votes
8answers
14k views

What is $\gcd(0,0)$?

What is the greatest common divisor of $0$ and $0$? On the one hand, Wolfram Alpha says that it is $0$; on the other hand, it also claims that $100$ divides $0$, so $100$ should be a greater common ...
56
votes
9answers
14k views

Has there ever been an application of dividing by $0$?

Regarding the expression $a/0$, according to Wikipedia: In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by $0$, gives $a$ (assuming $a\not= 0$), and ...
52
votes
2answers
2k views

Fractals using just modulo operation

Let us calculate the remainder after division of $27$ by $10$. $27 \equiv 7 \pmod{10}$ We have $7$. So let's calculate the remainder after divison of $27$ by $7$. $ 27 \equiv 6 \pmod{7}$ Ok, so ...
50
votes
4answers
3k 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 $...
49
votes
5answers
6k views

What is the smallest positive multiple of 450 whose digits are all zeroes and ones?

What is the smallest positive multiple of 450 whose digits are all zeroes and ones? I tried guess and check but the numbers grew big fast. Thanks in advance!
44
votes
3answers
2k views

How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? [duplicate]

Possible Duplicate: Highest power of a prime $p$ dividing $N!$ How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
42
votes
9answers
4k views

Find all positive integers $n$ s.t. $3^n + 5^n$ is divisible by $n^2 - 1$

As is the question in the title, I am wishing to find all positive integers $n$ such that $3^n + 5^n$ is divisible by $n^2 - 1$. I have so far shown both expressions are divisible by $8$ for odd $n\...
40
votes
6answers
7k views

If a prime number is reversed, and then appended to itself, why is the result always a composite number?

$2 \Rightarrow 22$ which is a composite number. $37 \Rightarrow 3773$ which is a composite number. $523 \Rightarrow 523325$ which is a composite number. $8123 \Rightarrow 81233218$ which is a ...
37
votes
11answers
10k views

How to prove that all odd powers of two add one are multiples of three

For example \begin{align} 2^5 + 1 &= 33\\ 2^{11} + 1 &= 2049\ \text{(dividing by $3$ gives $683$)} \end{align} I know that $2^{61}- 1$ is a prime number, but how do I prove that $2^{61}+1$ ...
37
votes
3answers
1k views

Prove that there exist infinitely many integers $(n^{2015}+1)\mid n!$

I conjecture that there exist infinitely many integers $n$ such that $$(n^{2015}+1)\mid n!.$$ I have seen a simpler problem that there exist infinitely many integers $n$ such that $(n^2+1)\mid n!$. ...
35
votes
12answers
15k views

Division of Factorials

I have a partition of a positive integer $(p)$. How can I prove that the factorial of $p$ can always be divided by the product of the factorials of the parts? As a quick example $\frac{9!}{(2!3!4!)} =...
35
votes
8answers
13k views

Why is $a^n - b^n$ 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 $a^n$ $-$...
33
votes
5answers
2k 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 $...
31
votes
7answers
4k views

Why do some divisibility rules work only in base 10?

Aside from the zero rule (a number ending with zero means it is divisible by the base number and its factors), why is it that other rules cannot apply in different bases? In particular for 3 one can ...
30
votes
8answers
13k views

Why $\gcd(qb+r,b)=\gcd(b,r)$?

Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so? Addendum by LePressentiment on 11/29/2013: (in the interest of http://meta.math....
30
votes
8answers
2k views

Prove that $(mn)!$ is divisible by $(n!)\cdot(m!)^n$

Prove that $$(n!)\cdot(m!)^n|(mn)!$$ I can prove it using Legendre's Formula, but I have to use the lemma that $$ \dfrac{\displaystyle\left(\sum_{i=1}^na_i\right)!}{\displaystyle\prod_{i=...
30
votes
1answer
3k views

Can Mickey Mouse divide by $7$?

In the figure displayed in the image below : To find the remainder on dividing a number by $7$, start at node $0$, for each digit $D$ of the number, move along $D$ black arrows (for digit $0$ do not ...
30
votes
2answers
2k views

When is $\binom{n}{k}$ divisible by $n$?

Is there any way of determining if $\binom{n}{k} \equiv 0\pmod{n}$. Note that I am aware of the case when $n =p$ a prime. Other than that there does not seem to be any sort of pattern (I checked up ...
30
votes
1answer
429 views

A spiralling sequence based on integer divisors. Has anyone noticed this before?

Firstly, please excuse the informal style of my explanation, as I am not a mathematician, although I am aware that this can be explained in more formal terms. I have mapped integers to points on a ...
29
votes
7answers
31k views

The product of $n$ consecutive integers is divisible by $n$ factorial

How can we prove that the product of $n$ consecutive integers is divisible by $n$ factorial? Note: In this subsequent question and the comments here the OP has clarified that he seeks a proof that "...
29
votes
5answers
8k views

Fibonacci modular results $\ F_n\mid F_{kn},\,$ $\, \gcd(F_n,F_m) = F_{\gcd(n,m)}$

Can any one give a generalization of the following properties in a single proof? I have checked the results, which I have given below by trial and error method. I am looking for a general proof, which ...
27
votes
8answers
33k 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 why ...
27
votes
17answers
49k views

How to Prove the divisibility rule for $3$

The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three. This is quite helpful for determining if really large ...
24
votes
6answers
14k views

If $n\ne 4$ is composite, then $n$ divides $(n-1)!$.

I have a proof and need some feedback. It seems really obvious that the statement is true but it is always the obvious ones that are a little trickier to prove. So I would appreciate any feedback. ...
24
votes
2answers
3k views

Elementary proof of Zsigmondy's theorem

I've been writing a not-so-short introduction to elementary number theory, supplying proofs for all theorems. When coming across Zsigmondy's theorem, it seemed difficult to find a proof available on ...
24
votes
2answers
243 views

An enigmatic pattern in division graphs

Draw the numbers $1,2,\dots,N$ on a circle and draw a line from $n$ to $m>n$ when $n$ divides $m$: For larger $N$ some kind of stable structure emerges which remains perfectly in place for ever ...
23
votes
10answers
2k 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 ...
23
votes
5answers
45k views

What is vector division?

My question is: We have addition, subtraction and muliplication of vectors. Why cannot we define vector division? What is division of vectors?
23
votes
1answer
771 views

One of any consecutive integers is coprime to the rest

After reading this question, I conjectured a generalization of it. Conjecture: Fix $k\in \mathbb N$. Then, for all $n\in \mathbb N$, one of $n+1,\ldots,n+k$ is coprime to the rest. I ...
21
votes
5answers
13k views

Why are all non-prime numbers divisible by a prime number?

In Euclid's infinite prime numbers proof, the logic is as follows: Assume a set $S$ of all prime numbers in existence is finite (there are a finite amount of primes) Then there must be a greatest ...
20
votes
4answers
2k 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 ...
20
votes
7answers
19k views

How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$?

I can see that this works for any integer $n$, but I can't figure out why this works, or why the number $42$ has this property.
20
votes
3answers
3k views

Prove that there exists a number divisible by 1999 with digit sum 1999

My nephew in the secondary school asked me how to solve the problem as stated in the title. Honestly, I do not have any idea how to do it: Prove that there exists a positive integer number such ...
20
votes
3answers
2k views

How to solve this algorithmic math olympiad problem?

So, today we had a local contest in my state to find eligible people for the international math olympiad "IMO" ... I was stuck with this very interesting algorithmic problem: Let $n$ be a natural ...
20
votes
4answers
620 views

Do there exist infinitely many pairs of primes $(p,q)$ such that $pq$ divides $2^{p-1}+2^{q-1}-2$?

A mathematician friend gave me this question (partly as a joke) a few months ago and it has puzzled me for a long time:- Do there exist infinitely many pairs of primes $(p,q)$ such that $pq|2^{p-1}+...
20
votes
0answers
478 views

Have I discovered an analytic function allowing quick factorization?

So I have this apparently smooth, parametrized function: The function has a single parameter $ m $ and approaches infinity at every $x$ that divides $m$. It is then defined for real $x$ apart from ...
20
votes
0answers
492 views

Dividing the whole into a minimal amount of parts to equally distribute it between different groups.

Suppose we have a finite amount of numbers $x_1, x_2, ..., x_n$ ($x_i\in\mathbb{N}$) and an object that should be divided into parts in such a way that it can be without further dividing distributed ...
19
votes
7answers
2k views

How can I find integers which satisfy $\frac{150+n}{15+n}=m$?

Here are some facts about myself: In 2017, I was $15$ years old. Canada, my country, was $150$ years old. When will be the next time that my country's age will be a multiple of mine? I've toned ...
19
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 ...
19
votes
4answers
2k 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 ...