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

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9
votes
6answers
259 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 ...
6
votes
3answers
12k views

Find the largest prime factor

I just "solved" the third Project Euler problem: The prime factors of 13195 are 5, 7, 13 and 29. What is the largest prime factor of the number 600851475143 ? With this on Mathematica: ...
4
votes
5answers
164 views

If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$ [duplicate]

Prove or disprove 'If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$, with $a,b\not= 0$' I need to prove this statement. I think it is true and also the converse is true. I took some examples such as ...
3
votes
5answers
121 views

How to show $n(n+1)(2n+1) \equiv 0 \pmod 6$?

I've been asked to show that: $n(n+1)(2n+1) \equiv 0 \pmod 6$ I found in a previous question that: $n(n+1)$ was divisible by $2$ and resulted in an even number e.g $n(n+1) \equiv 0 \pmod 2$ so I ...
3
votes
2answers
3k views

Proof of $\gcd(a,b)=ax+by$

Here is my proof of $\gcd(a,b)=ax+by$ for $a, b, x, y \in \mathbb{Z}$. Am I doing something wrong? Are there easier proofs? $a,b \in \mathbb{Z}, g=\gcd(a,b)$ and suppose $g \neq ax + by$. Let $c$ be ...
2
votes
3answers
124 views

Prove that distinct Fermat Numbers are relatively prime [duplicate]

The Fermat numbers are defined by $F_m = 2^{2^m} + 1$. Prove that for $m \ne n$ we have $(F_m, F_n) = 1$. I have to first prove that $F_{m+1} = F_0F_1 \cdots F_m + 2$ by representing $F_{m+1}$ in ...
2
votes
2answers
92 views

Find all primes of the form $2^{2^n} + 5$ for a nonnegative integer n

I'm a little lost on how to do this problem. It looks a lot like the definition for the Fermat numbers: $F_n = 2^{2^n} + 1$, however I'm not sure how to use that in order to find all of the primes of ...
2
votes
2answers
133 views

Asymptotic divisor function / primorials

Let $p_n\#\equiv\prod_{k=1}^{n}p_k$ (primorial), and $\sigma(n)=\sum_{d|n}^{}d$ (divisor function). Could someone please tell me what the general asymptotic of $\dfrac{\sigma(p_n\#)}{p_n\#}$ is? It ...
2
votes
2answers
105 views

Extended Euclidean Algorithm problem

I'm confused about how to do the extended algorithm. For example, here's the gcd part gcd(8000,7001) $$\begin{align}8000 &= 7001\cdot1 + 999\\ 7001&=999\cdot 7+8\\ 999&=8\cdot 124+7\\ ...
2
votes
4answers
157 views

Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.

Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$. I tried to set $\gcd(a, b)$ to $b$ and used the fundamental theorem of arithmetic to prove that it is divisible by $a$, but I ...
2
votes
3answers
99 views

System of two simple modular equations

$$x \equiv -7 \mod 13$$ $$x \equiv 39 \mod 15$$ I need to find the smallest x for which these equations can be solved. I've been always doing this using Chinese Reminder Theorem, but it seems that it ...
1
vote
3answers
50 views

If $\gcd (a,b) = 1$, what can be said about $\gcd (a+b,a-b)$?

Suppose $a, b \in \mathbb{Z}$, $a > b$, and $\gcd (a,b) = 1$. What can be said about $\gcd (a+b,a-b)$? Is it true in general that $\gcd (a+b,a-b) \leq 2$?
1
vote
2answers
94 views

Proving a Pellian connection in the divisibility condition $(a^2+b^2+1) \mid 2(2ab+1)$

I'm trying to prove that all integer solutions $a > b \ge 0$ to the divisibility condition in the title, namely $$(a^2+b^2+1) \mid 2(2ab+1),$$ are given by ...
1
vote
4answers
107 views

How to solve the equation $n^2 \equiv 0 \pmod{584}$?

Well, I've confused when trying to solve this equation can anybody help me : $n^2 \equiv 0 \pmod{584}$ I tried to factorize the $584$ i got $584=2^3\times73$. so $n^2$ has to be divisible by $2^3$ ...
1
vote
2answers
270 views

Rabin's test for polynomial irreducibility over $\mathbb{F}_2$

I know that $f(x) = x^{169}+x^2+1$ is a reducible polynomial over $\mathbb{F}_2$. I want to show this using Rabin's irreduciblity test. First, I then have to check if $f$ is a divisor of ...
1
vote
4answers
170 views

$(a^{n},b^{n})=(a,b)^{n}$ and $[a^{n},b^{n}]=[a,b]^{n}$?

How to show that $$(a^{n},b^{n})=(a,b)^{n}$$ and $$[a^{n},b^{n}]=[a,b]^{n}$$ without using modular arithmetic? Seems to have very interesting applications.$$$$Try: $(a^{n},b^{n})=d\Longrightarrow ...
1
vote
4answers
2k views

Prove that if a, b, x, y are integers with ax + by = gcd(a, b) then gcd(x,y)= 1

How would I go about proving this: Prove that if a, b, x, y are integers with ax + by = gcd(a, b) then gcd(x,y)= 1
0
votes
1answer
65 views

Parity of number of factors up to a bound?

Consider $b,n\in\mathbb{N}$ where $b\leq n$. We want to find the parity (ie. odd or even) of the number of divisors of $n$ that are $\leq b$. The question is to find a fast algorithm to find that ...
6
votes
3answers
125 views

Why if $n \mid m$, then $(a^n-1) \mid (a^m-1)$?

My Number Theory book says that for $n, m$ be positive integers and $a>1$, then $(a^n -1)\mid(a^m -1)$ if and only if $n\mid m$. I understand the proof for only if part, but in if part the ...
5
votes
5answers
768 views

Is it possible to get 1/3 without dividing by 3?

So I need to divide a rectangle into 3 equals parts, but without fractions. It's one of those old "You have two jars of two sizes and need to get an exact amount of some other size" type problems, ...
4
votes
2answers
94 views

A non-UFD such that $a^2 \mid b^2$ does not lead to $a\mid b$

Is there any non-UFD that is a commutative ring such that $a^2 \mid b^2$ does not always lead to $a\mid b$? It would be preferable if examples are something that does not involve ...
4
votes
6answers
137 views

Proving $\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$

How would you go about proving that $$\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$$ for any two integers $a$ and $b$? Intuitively it is true because when you divide $a$ and $b$ by ...
4
votes
5answers
554 views

Prove 24 divides $u^3-u$ for all odd natural numbers $u$

At our college, a professor told us to prove by a semi-formal demonstration (without complete induction): For every odd natural: $24\mid(u^3-u)$ He said that that example was taken from a high ...
4
votes
4answers
269 views

If $b^2$ is the largest square divisor of $n$ and $a^2 \mid n$, then $a \mid b$.

I am trying to prove this: $n$, $a$ and $b$ are positive integers. If $b^2$ is the largest square divisor of $n$ and $a^2 \mid n$, then $a \mid b$. I want to prove this by contradiction, and I ...
3
votes
3answers
238 views

A polynomial with integer coefficient

I'm struggling with this question: Suppose $P(x)$ is a polynomial with integer coefficients such that non of the values $P(1),...,P(2010)$ is divisible by $2010$. Prove that $P(n)\neq 0$ for all ...
3
votes
1answer
347 views

The product of any three consecutive natural numbers is divisible by 9.prove or find a counterexample

can anyone give me feedback on my solution false counterexample: n(n+1)(n+2) let n= 1 1(2)(3) =6 is not divisible by 9 is this correct?
3
votes
1answer
115 views

Find two elements that don't have a gcd in a subring of Gaussian integers

Find two elements in the domain $R := \{ x + 2y \sqrt {-1} \mid x,y \in \mathbb{Z} \}$ that do not have a gcd. I have no idea how to start. But I know if we consider $R^\prime = \{ x + y \sqrt ...
3
votes
6answers
222 views

Proof that $a^5 b - b^5 a$ is divisible by $30$ for any integers $a$ and $b$

I am trying to prove that $a^5\times b - b^5\times a$ is divisible by $3$. The actual task is to prove divisibility by $30$ but I have managed to prove that the expression is divisible by $5$ and $2$. ...
3
votes
1answer
253 views

GCD of Fibonacci-like recurrence relation

What is the greatest common denominator of $t(c^a)$ and $t(c^b)$, if $t(n) := k_1 f_1^n + k_2 f_2^n $? I already found out that the gcd is always a member of $t(n), n \in N $. $t(n)$ was originally ...
3
votes
1answer
85 views

Four fractions of certain factorials give another factorial

Let $n>0$ and $s_n=\sum_{k=1}^n k$. I looked at the expressions $\displaystyle\frac{s_n!}{(s_n-n)!}$ and found that the fraction is another factorial for $k=1,2,3,4$, i.e. ...
3
votes
1answer
1k views

Prove the converse of Wilson's Theorem

... namely that if $n > 1$ and $(n − 1)!\equiv−1\pmod{n}$, then $n$ is prime. This is for a number theory class I'm in at Penn State. My idea is to follow accordingly, but I can't get it ...
3
votes
5answers
142 views

Prove that if $6 \mid (2a + 4)$ and $9 \mid (12 + 3b)$ then $3 \mid (a + b)$

Could you help me with the problem below? Prove that if $6 \mid (2a + 4)$ and $9 \mid (12 + 3b)$ then $3 \mid (a + b)$. Thank you!
3
votes
2answers
391 views

Prove that every integer $n>0$ with $\gcd(n,10) = 1$ has a multiple that can be written with only the digit $9$. [duplicate]

Possible Duplicate: Proof that a natural number multiplied by some integer results in a number with only one and zero as digits The question is as stated. I'm really stumped on it. It seems ...
3
votes
2answers
331 views

Number of Divisor

How to find the Number of divisors of a number 'n' that are also divisible by another number 'k' without looping through all the divisors of n? I tried the following: Stored powers of all prime ...
2
votes
1answer
114 views

Do these arithmetic rules work? They extend the number system by a zero not based on the empty set that is a divisor with unique quotients.

These rules are part of an attempt to define an additive identity in terms of division in basic standard arithmetic. The difficulties with defining division by $0$ are well known. In order to ...
2
votes
1answer
43 views

The greatest common divisor is the smallest positive linear combination

How to prove the following theorems about gcd? Theorem 1: Let $a$ and $b$ be nonzero integers. Then the smallest positive linear combination of $a$ and $b$ is a common divisor of $a$ and $b$. ...
2
votes
0answers
76 views

Find Gcd summation fast?

Find the value of the summation: $$ val=\left( \sum_{i=1}^a \sum_{j=1}^b\sum_{k=1}^c....\sum_{x=1}^p GCD(i,j,k,..x) \right)$$ Contraints $2\leq$number of summation terms$\leq 500$, $1\leq ...
2
votes
3answers
142 views

The number $n^4 + 4$ is never prime for $n>1$

I am taking a basic algebra course, and one of the proposed problems asks to prove that $n^4 + 4$ is never a prime number for $n>1$. I am able to prove it in some particular cases, but I am not ...
2
votes
1answer
53 views

Values of $gcd(a-b,\frac{a^p-b^p}{a-b} )$

I don't know how to prove the following result. Let $p$ be a prime number and let $a,b \in \mathbb Z$ such that $gcd(a,b)=1$ Then $gcd(a-b,\frac{a^p-b^p}{a-b}) = 0 $ or $ p $ I know that ...
1
vote
0answers
147 views

Vieta jumping with non-monic polynomials

I have recently discovered Vieta jumping as a problem-solving technique. In order to teach myself about it, I have located most (all of?) the standard references, both here on MSE and "out there" (via ...
1
vote
2answers
58 views

If I remove the premise $a\neq b$ in this question, will the statement still be true?

I encountered this proving problem, I can do the proof but my question is why in the condition/premise we need $a$ to be unequal to $b$? My guess is that even $a=b$, the statement is still true, is it ...
1
vote
3answers
101 views

Divisibility test for $4$

Claim: A number is divisible by $4$ if and only if the number formed by the last two digits is divisible by $4$. Here's where I've gotten so far. Let $x$ be an $(n+1)$-digit number. So $x= ...
1
vote
2answers
118 views

Let $d$ be a positive odd integer. Prove that there exists a positive $n \le d$ such that $d | 2^n − 1$.

I'm stuck on this question from my textbook which doesn't even have a solution. Any ideas ?. Help would be much appreciated. $$ d\, \left\vert\right.\, \left(2^{n} − 1\right) $$
1
vote
3answers
160 views

Basic Modulo Question

I've been having trouble with this example while studying for my exams. Why is $$2023^{2297}\equiv 20 \pmod{3953}\;?$$ Thanks so much for any help I can get! The examples solves the answer by ...
0
votes
2answers
61 views

Using inclusion-exclusion principle to count the integers in $\{1, 2, 3, \dots , 100\}$ that are not divisible by $2$, $3$ or $5$

Question: How many integers in $\{1, 2, 3, \dots , 100\}$ are not divisible by $2$, $3$ or $5$? Can anyone give me a full explain of how this applies to the inclusion-exclusin principle? Because I ...
0
votes
1answer
60 views

Show that the number 9 divides the number $ m$ if and only if the sum of the digits of the number $ m $ is divisible by 9. [duplicate]

Show that the number 9 divides the number $ m$ if and only if the sum of the digits of the number $ m $ is divisible by 9. Show that the number 3 divides the number $ m$ if and only if the sum of the ...
0
votes
3answers
119 views

How to prove if $n$ is prime and $n | a^2$ then $n | a$?

My professor assigned this for homework but I don't understand how to connect the dots. He suggested using the fact that $\gcd (x,y) \cdot \operatorname{lcm} (x,y) = xy$ but I'm not sure how that's ...
0
votes
3answers
267 views

Prove that if a and b are integers, then there are unique integers q and r such that $a = bq + r$, $-|b|/2 < r \le |b|/2$ [closed]

Prove that if a and b are integers, then there are unique integers q and r such that $$a = bq + r,$$ with the restriction that$$-|b|/2 < r \le |b|/2$$
0
votes
1answer
98 views

Two issues of Number Theory

Knowing in Fibonacci sequence$$u_n\mid u_m\Longleftrightarrow n\mid m$$ Question 1: In Fibonacci sequence, show that $$5\mid u_m\Longleftrightarrow 5\mid m$$ Show: $\Longrightarrow$ In ...
0
votes
1answer
84 views

Prime divisibility in a prime square bandtwidth

I am seeking your support for proving (or fail) formally the following homework: Let $p_j\in\Bbb P$ a prime, then any $q\in\Bbb N$ within the interval $p_j<q<p_j^2$ is prime, if and only if ...