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

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3
votes
8answers
118 views

Proving that $12^n + 2(5^{n-1})$ is a multiple of 7 for $n\geq 1$ by induction

Prove by induction that $12^n + 2(5^{n-1})$ is a multiple of $7$. Here's where I am right now: Assume $n= k $ is correct: $$12^k+2(5^{k-1}) = 7k.$$ Let $n= k+1 $: $$12^{k+1} + 2(5^k)$$ ...
6
votes
6answers
146 views

How to prove that $2^{n+2}+3^{2n+1}$ is divisible by 7 using induction?

I want to prove that $2^{n+2}+3^{2n+1}$ is divisible by 7 using induction. My first step is replace $n$ with $1$. $2^{1+2}+3^{2(1)+1}$ $2^3+3^3$ $8+27$ $35 = 7\times 5$ The next step is assume ...
3
votes
4answers
92 views

Number of fingers of a Martian

I have a question about what seems to be modular arithmetic, but I can't quite get the answer. The problem goes along the lines of: It is often said Earthlings use the decimal system because they ...
4
votes
0answers
68 views

$d$ and $d+1$ both dividing certain integers

\begin{align} & 1\cdot 72 \\ & 2\cdot 36 \\ & 3\cdot 24 \\ & 4\cdot 18 \\ & 6\cdot 12 \\ & 8\cdot 9 \end{align} When the divisors of a number are listed in this way, let us ...
1
vote
0answers
36 views

Bezout Coefficient for polynomials in sage?

I want to find the bezout coefficient for those 2 polynomials : $f = 1+x-x^2-x^4+x^5$ and $g = -1+x^2+x^3-x^6$ when I use the gcd function in sage the output is : ...
0
votes
1answer
43 views

Are Zero Degree polynomials Considered monics?

DO zero degree polynomials that is constant polynomials considered monic polynomials? Example F(x)=16 Does it Matter the Field or the Integral region where i take the coeficients from?Sorry if the ...
6
votes
2answers
131 views

Prove or disprove : $a^3\mid b^2 \Rightarrow a\mid b$

I think it's true, because I can't see counterexamples. Here's a proof that I am not sure of: Let $p_1,p_2,\ldots, p_n$ be the prime factors of $a$ or $b$ \begin{eqnarray} a&=& ...
11
votes
2answers
2k views

$\gcd(b^x - 1, b^y - 1, b^ z- 1,…) = b^{\gcd(x, y, z,…)} -1$ [duplicate]

Possible Duplicate: Number theory proving question? Dear friends, Since $b$, $x$, $y$, $z$, $\ldots$ are integers greater than 1, how can we prove that $$ \gcd (b ^ x - 1, b ^ y - 1, b ^ ...
1
vote
1answer
25 views

Is it possible to compute $\sigma(AB)$ if $\gcd(A, B) = C > 1$?

Let $\sigma$ be the classical sum-of-divisors function. I know, for one, that $\sigma$ is weakly multiplicative (that is, $\sigma(xy) = \sigma(x)\sigma(y)$ whenever $\gcd(x, y) = 1$). Well, of ...
1
vote
1answer
32 views

A question about $\gcd$ and divisibility

Let $\sigma$ be the classical sum-of-divisors function. Suppose that I have the following equations: $$2n^2 - \sigma(n^2) = \frac{\sigma(n^2)}{q^k}\cdot{\sigma(q^{k-1})}$$ $$2n^2 - \sigma(n^2) = ...
3
votes
1answer
29 views

What is $\gcd(\sigma(q^{k-1}), \sigma(q^k))$?

Let $\sigma$ denote the classical sum-of-divisors function. What is $\gcd(\sigma(q^{k-1}), \sigma(q^k))$? Update: I have transferred the transcript of my attempt to an actual answer to this MSE ...
0
votes
2answers
55 views

Is there a solution to this system of equations?

Is there an integer solution to this system of equations? $$\gcd(x, \sigma(x)) = 2x - \sigma(x) = \frac{x}{3} = \frac{\sigma(x)}{5}$$
0
votes
1answer
57 views

Prove: $\gcd(n^a-1,n^b-1)=n^{\gcd(a,b)}-1$ [duplicate]

I'm trying to prove the following statement: $$\forall_{a,b\in\Bbb{N^{+}}}\gcd(n^a-1,n^b-1)=n^{\gcd(a,b)}-1$$ As for now I managed to prove that $n^{\gcd(a,b)}-1$ divdes $n^a-1$ and $n^b-1$: Without ...
6
votes
4answers
98 views

Divisibility of polynomials in a subfield of a field.

I am trying to prove the following assertion: Let $K\subset L$ be fields, let $f,g\in K[x]$ be such that $f\mid g $ in $L[x]$, then $f\mid g$ in $K[x]$. We clearly have that $fh=g$ for some ...
1
vote
3answers
277 views

Why does Wolfram Alpha say that $n/0$ is complex infinity?

I typed a number divided by 0 on Wolfram Alpha and thought that it would say "undefined". However, when I pressed enter it told me that the answer is complex infinity. I have always been taught ...
0
votes
0answers
29 views

When is ${(1-t^2)^{-N/2}}{\det(f_t(A))}$ expressible as polynomial?

Given a matrix valued function $f_t:\mathbb R^{N\times N} \mapsto \mathbb R^{N\times N}$. For $f_t(A)=B$ both $A$ and $B$ are symmetric. Which properties can be assigned to $f_t$, when the following ...
1
vote
2answers
36 views

Proving that $p^{\alpha + \beta + 1} \mid {n \choose k} p^{k\alpha}$ when $p^\beta \mid n$.

Let $n,\alpha\in\mathbb{N},\beta\in\mathbb{N}_0$, and let $p$ be odd prime number s.t. $p^\beta|n$. How do we prove that $p^{\alpha+\beta+1}|{n\choose k}p^{k\alpha}$ for every ...
2
votes
2answers
45 views

Prove that $12 \mid m \iff$ both $6 \mid m$ and $4 \mid m$.

Give a formal proof to the following theorem which I do not know where to start. Theorem: For all natural numbers 'm', 12 divides m only if 6 divides m and 4 divides m.
1
vote
7answers
56 views

Direct Proof on Divisibilty

Using Induction proof makes sense to me and know how to do, but I am having a problem in using a direct proof for practice problem that was given to us. The problem is: For all natural numbers $n$, ...
1
vote
2answers
55 views

prove $(m) \subset (n)$ iif $n$ divides $m$

For non-zero integers $m$ and $n$, prove $(m) \subset (n)$ iif $n$ divides $m$, where $(n)$ is the principal ideal. My attempt is following. For non-zero integers $m$ and $n$, assume that $(m) ...
4
votes
2answers
403 views

Fastest way to find if a given number is prime

Given a random number, what would be the quickest possible way of finding out whether it was prime? Obviously, one could just iterate through the number in order to see if it was divisible by ...
0
votes
1answer
136 views

Division rules for other number systems? [duplicate]

How could we make the same division rules for other number systems, like in our decimal system: a number is divisible with 2 if it's last digit is 0,2,4,6,8, by 3 if the sum of digits is divisible ...
9
votes
2answers
3k 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 ...
2
votes
1answer
83 views

Divisibility by 9 with negative number

I know the rule to check divisibility by 9: check if the sum of the digits of the number is divisible by 9. But what if the number is negative? Thanks in advance!
5
votes
1answer
33 views

$x-1$ in base $x$ counting systems

Please excuse the lack of expertise. I'm not a mathematician, nor have I studied it since high school. I was thinking about how all the digits of multiples of $9$ summed equal a multiple of $9$. I ...
5
votes
8answers
166 views

Prove with induction that $11$ divides $10^{2n}-1$ for all natural numbers.

$$10^{2(k+1)}-1 = 10^{2k+2}-1=10^{2k}\cdot10^{2}-1$$ I feel like there's something in that last part that should make it work, but I can't grasp it. Am I missing something obvious? Am I going in the ...
3
votes
1answer
85 views

Solving a Diophantine equation3

The Diophantine equation that I have to solve is: $$343x^2-27y^2=1$$ This question has already been posted by other user but it has not received an answer. I proved to solve it. This is my attempt: ...
6
votes
2answers
76 views

Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$

Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$. This is what I have done so far: Let $d = \gcd(a,b)$. Then $d=ax+by$ for some $x,y$. Then $d^2 =(ax+by)^2 = a^2x^2 + 2axby+b^2y^2$. I am trying to create a ...
1
vote
3answers
61 views

How to find the remainder of polynomial division?

Im trying to solve this problem but I do not understand what the question is asking: Let $n\ge 2$ be an integer and $ p_n(x) $ be the polynomial: $$ p_n(x) = (x-1)+(x-2)+\cdots+(x-n) $$ What is the ...
0
votes
1answer
22 views

Mysterious divisibility condition showing up in computation of determinant of certain sparse matrices

Notation: by the $d$'th diagonal of an $n \times n$ matrix $A$ I will denote the diagonal parallel to the main diagonal that starts in row 1, column $d$. I will extend this definition in the obvious ...
1
vote
3answers
67 views

Divisibility number theory problem

How many $k,m$ exist such that $ \frac {k^2+m^2}{2(k-m)}$ is also an integer. $k,m \in \mathbb {Z} ^ + $ My guess that there is finitely many solutions but I can't seem to be able to prove so.
5
votes
5answers
90 views

What is the biggest $n$ in $4^n$ that divides $7^{2048} - 1$?

A few days ago I stumbled on the following question, it was used in the Museum of mathematics masters tournament: What is the biggest integer $n$ in $4^n$, that divides $7^{2048} - 1$? a) 1 b) 3 ...
23
votes
7answers
4k 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 $a^n$ ...
-1
votes
1answer
44 views

Let p and q be two different prime numbers [closed]

a) Let p and q be two different prime numbers. If $p+q^2│p^2+q$, prove that $p+q^2│pq-1$ b) Find all prime numbers p such that $p+121│p^2+11$
1
vote
0answers
51 views

Multiple of power of 5 with only the digits 2,5,6

after helping a friend solve a homework, I asked myself the following question: $H\subseteq\{1,2,\ldots,9\}$, $T(H)=\{n\in\mathbb{N}:$ all the digits in the decimal representation in $n$ belong to ...
1
vote
1answer
14 views

quadratic form polynomial divisibility vs. matrix pointwise multiplication.

Given matrix $V',W',Y'$ is of $d\times m (d\le m)$ ; column vector $c$ is of size $m$; $r_i, i=1,...,d$ are distinct; and each row of the matrix A is $A_i=(r_i^0 ... r_i^{d-1})$. So, A is of $d\times ...
1
vote
1answer
23 views

Given $2^n$, what is the largest power of $2$ that will divide any random concatenation of base $10$ digits of powers of $2$ ending with $2^n$?

My first thought was that it would be $2^n$ itself, for example, if you concatenate $4$ and $2$ to get $42$, that's divisible by $2$ but not by $4$. But whit $2^9 = 512$, you can concatenate $16$ and ...
3
votes
2answers
50 views

How to prove that if $m$ is squarefree, then $d^2 \lvert mb^2 \implies d \lvert b$

This statement was given in my number theory textbook when analyzing quadratic fields, and I am not seeing how to prove it. $m$ is a squarefree (not divisible by the square of any number) integer and ...
1
vote
4answers
71 views

Dividing by Zero Using Exponents

I am sorry if I am missing some fundamental rule disproving my question, but I am very confused here. So $0^0 = 1$ and $x/x = x^0 = 1$ so if $x = 0$ $0/0 = 1$ The problem here is people have ...
4
votes
2answers
91 views

Find all values of $x,y,z$ positive integers such that $4^x+4^y+4^z$ is a perfect square

I have to solve the equation $$4^x+4^y+4^z=k^2$$ I posted my solution but i don't know if there are other solution. How can i demonstrate that this expression is a perfect square? Are there oter ...
1
vote
1answer
74 views

The following is a necessary condition for a number to be prime, from its digit expansion. Has it been referred somewhere?

Concerning a numbers’ digits we know some necessary conditions on them for the number to be prime, besides the last digit having to be odd (except for prime 2). For example in decimal representation ...
4
votes
3answers
455 views

Prove that the integers $x$, $x+6$, $x+12$, $x+18$, $x+24$ can only be prime if $x$ is $5$.

Prove that the integers $x$, $x+6$, $x+12$, $x+18$, $x+24$ can only be prime if $x$ is $5$. I am very new to proofs and not completely sure of how to approach this one. I tried several different ...
1
vote
2answers
29 views

if d divides n then prove that fibonacci of d divides fibonacci of n

prove that if $d$ divides $n$ then prove that fibonacci of $d$ divides fibonacci of $n$. i have tried to write $F(n)$ as a multiple of $F(d)$ using the fact that $n = ad$ for some natural $a$ but got ...
4
votes
4answers
216 views

Odd divisibility induction proof

Prove that for odd $n>3$ $$64\ | \ n^4-18n^2+17$$ I checked that for $n=5$ it works. I think I need to assume that for $2n+1$ it holds and show that $2n+3$ also holds. Any ideas?
2
votes
1answer
30 views

Find remainder when $20^{13}$ is divided by $4940$

Find remainder when $20^{13}$ is divided by $4940$ I have solved this, but am hoping for an elegant solution. My solution: $r(20^{13}||4940) = 20 \times r(20^{12}||247)$ where $r(a||b)$ is the ...
3
votes
2answers
52 views

Prove that if $a \mid n$ then $a^2\mid (n + 1)(n − 1) + 1$

I have this review question for an exam and I was hoping someone can help me solve it: Prove that if $a \mid n$ then $a^2\mid (n + 1)(n − 1) + 1$ this is what I have so far, not sure if it is ...
1
vote
4answers
74 views

Let $a,b$ be relative integers such that $2a+3b$ is divisible by $11$. Prove that $a^2-5b^2$ is also divisible by $11$.

The divisibility for $11$ of $a^2 - 5b^2$ can be easily verified; in fact: $$a \equiv \frac {-3}{2}b \pmod {11}$$ therefore $$\frac {9}{4}\cdot b^2 - 5b^2 = 11(-\frac{b^2}{4}) \equiv 0 \pmod {11}.$$ ...
1
vote
9answers
81 views

Find integers $m$ and $n$ such that $14m+13n=7$.

The Problem: Find integers $m$ and $n$ such that $14m+13n=7$. Where I Am: I understand how to do this problem when the number on the RHS is $1$, and I understand how to get solutions for $m$ and ...
1
vote
1answer
88 views

Fermat's Little Theorem and prime divisors

Let $a,b\in\Bbb N$ and $a+b$ be an even number. Assume $a^2 - b^2 - a$ is an exact square, say $c^2$. Let $m = \frac {a+b}2$ and $n = \frac {a-b}2$. Then, $$(4m-1)(4n-1) = 4(4mn-m-n) + 1 = ...