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

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4
votes
3answers
166 views

What is so great about 7?

I'm going to write down my problem verbatim: Write down the integers from $1$ to $50$ in rows of $10$ numbers each. Mark out $1$, and then cross out all multiples of $2$ greater than $2$ ...
0
votes
3answers
25 views

Solving for mod indirectly

How many positive integers $n$ exist such that $\frac{680}{n}$ is an integer? So, this is quite obvious, $680 \equiv 0 \pmod{n}$ How should I solve for $n$? There will be multiple $n$?
2
votes
2answers
93 views

Factoring numbers of the form $11111111$

Why $11111111$ is divisible by $73$? How can we get all the prime factors? It is clear that it is divisible by $11$. Is there any formulae for $1111...11$ ($n$ times)? Give me some idea. Thanks in ...
0
votes
0answers
24 views

finding the logic behind the division method of hcf [closed]

How does the division method of finding hcf work.should we consider that their exist a common factor that divides both the numbers.
1
vote
1answer
61 views

Use division algorithm to prove for any odd integer n, $n^2 -1$ is a multiple of 8.

Here is what I know if n is any odd integer then $n$ can be expressed as $n=2k+1 ~~~ where~k\in\mathbb{Z}$.So $n^2-1=(2k+1)^2 -1=4k^2+4k=4k(k+1)$ but $k(k+1)~~ is~~even$. Thus $k(k+1)=2t, t\in ...
0
votes
0answers
28 views

>Find all pairs of positive integers $(m, n)$, so that $1 + x + x^2 +\ldots+x^m \mid 1 + x^n + x^{2n} +\ldots +x^{mn}$

Find all pairs of positive integers $(m, n)$, so that $1 + x + x^2 +\ldots+x^m \mid 1 + x^n + x^{2n} +\ldots +x^{mn}$ I have to find $(m, n)$ such that ...
2
votes
2answers
122 views

If GCD of a list of numbers is 1, is it a necessary condition that GCD of at least one pair of numbers from the list should be 1?

Suppose our numbers are {2, 6, 3}. GCD (2, 6, 3) = 1, GCD (2, 6) = 2, GCD (6, 3) = 3, but GCD(2, 3) = 1 If GCD(a,b,c) = p, GCD(a,b) = q, GCD(b,c) = r, GCD(c,a) = s, is it possible that p = 1 and (q ...
3
votes
2answers
73 views

Divisibility Of Positve Integers [closed]

Suppose a,b and c are three positive integers which satisfy the condition that ($a$2+$b$2+$c$2) is divisible by $(a+b+c)$. Prove that there exists infinitely many positive integers $n$ for which ...
0
votes
1answer
24 views

Divisibility of binomial coefficients

I have got this series of binomial coefficients - $${2n\choose 0}+3\times{2n\choose 2}+3^{2}\times{2n\choose 4}+\ldots +3^{n}\times{2n\choose 2n}$$ I have to prove this to be divisble by ...
2
votes
3answers
56 views

Proof of non divisibililty of $\binom{n}{r}$ with a prime $p$

I came across this : "It is possible to show that if $p$ is prime, $\binom{n}{r}$ is not divisible by $p$ if and only if the addition $r + (n-r)$, when written in base $p$, has no carries. This means ...
2
votes
2answers
162 views

7 digit number consisting of 7s and 5s

Find all the 7 digit numbers that have only 5 and 7 as their digits and divisible by both 5 and 7. I have no clue how to use divisibility of 7 to solve this problem. DO i need to check all the 64 ...
8
votes
3answers
182 views

centenes of $7^{999999}$

What is the value of the hundreds digit of the number $7^{999999}$? Equivalent to finding the value of $a$ for the congruence $$7^{999999}\equiv a\pmod{1000}$$
1
vote
2answers
32 views

GCD Using Euclidean Algorithm

How do I find the GCD of $65024$ and $128397$? And how do I express the GCD as a linear combination of $65024$ and $128397$ of the form $g = a\cdot 65024 + b\cdot 128397$? My work: $128397 = ...
1
vote
0answers
52 views

Evaluate $\sqrt{2^{2014} + 2^{2011} + 2^{2006}} \pmod{17}$

Evaluate: $$I = \sqrt{2^{2014} + 2^{2011} + 2^{2006}} \pmod{17}$$ $$I = \sqrt{2^{2006}\cdot (1 + 2^{5} + 2^{8} )} \pmod{17} = 2^{1003} \cdot \sqrt{2^8 + 2^5 + 1} \pmod{17}$$ The answer is $0$ ...
-1
votes
1answer
49 views

Solve $5991x + 289 \equiv 0 \pmod{2014}$

Solve: $$5991x + 289 \equiv 0 \pmod{2014}$$ $$5991x \equiv -289 \equiv 1725 \pmod{2014}$$ I need to find the inverse of $5991$ modulo $2014$. Start with Euclid's algorithm: $$5991 = 2(2014) + ...
0
votes
1answer
21 views

Two recurrence relations gcd proof

Let $q_1, q_2,\ldots$ be a sequence of integers with $q_i\gt0$ for all $i\gt 1$. Define $(a_n)_{n\ge-1}$ and $(b_n)_{n\ge-1}$ by the following recurrence relations: $a_{-1}=0,\ a_0=1,\ b_{-1}=1,\ ...
1
vote
1answer
35 views

Is there a set of integers where all differences are relatively prime?

Is there an infinite subset $\mathcal S\subset \mathbb Z$ with the property that for any 4-tuple of distinct elements $x,y,z,w\in \mathcal S$ $$ \gcd(x-y,z-w)=1? $$
4
votes
3answers
135 views

True or False: $2^{2^{2011}} \text{ divides } 2^{2^{2012} }$

True or false: $$2^{2^{2011}} \text{ divides } 2^{2^{2012} }$$ Give your justifications. I don't know how to start this problem so far. But, I guessed like this, $$2^{\underbrace{2\times ...
1
vote
3answers
50 views

If gcd$(a, 4) = 2$ and gcd$(b, 4) =2$, then gcd$(a + b, 4) = 4$ [closed]

If gcd$(a, 4) = 2$ and gcd$(b, 4) =2$, then gcd$(a + b, 4) = 4$ can someone help me solve this.
0
votes
1answer
99 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 ...
2
votes
2answers
67 views

Determining $\gcd(94, 27)$

I want to determine $\gcd(94, 27)$. Using the Euclidean algorithm, I got \begin{align} 94 &= 27 (3) + 13 \\ \implies 27 &= 13 (2) + 1 \\ \implies \;\;2 &= 2 (1) \end{align} Does this ...
0
votes
2answers
49 views

If $d$ divides $k$ and $d$ divides $n$, then $d$ divides $(8k - 3n)$

Suppose that $k$, $n$, and $d$ are integers and $d$ is not $0$. Prove: If $d$ divides $k$ and $d$ divides $n$, then $d$ divides $(8k - 3n)$. You may not use the theorem stating the following: Let ...
2
votes
3answers
78 views

The number of positive integers less than 1000 with an odd number of divisors

How many positive integers less than 1000 have an odd number of positive integer divisors? Well I know that the number has to be composite because a prime number has 2 divisors, which are 1 and ...
3
votes
2answers
95 views

prove that $2^n+2^{n-1}+2^{n-2}+8^n-8^{n-2}$ is a multiple of 7

Prove that a number $2^n+2^{n-1}+2^{n-2}+8^n-8^{n-2}$ is a multiple of 7 for every natural $n\ge2$. I am not sure how to start.
0
votes
3answers
18 views

Greatest common divisor of an integer 'a' and it's sum with 2.

I need to prove that the $\gcd(a, a+2)$ equals either 1 or 2. Intuitively this makes sense to me. If a is an odd integer then the gcd is 1, if a is even, the gcd is 2. I'm having trouble writing a ...
2
votes
5answers
634 views

Suppose that $5\leq q\leq p$ are both prime. Prove that $24|(p^2-q^2)$. [duplicate]

Suppose that $5\leq q\leq p$ are both prime. Prove that $24|(p^2-q^2)$. This is what I got so far. I figured that since $p,q$ are bigger than $5$, there are only odd primes for this conjecture. ...
3
votes
1answer
39 views

Way to show divisibility without using Euclid's lemma.

The generalized version of Euclid's lemma states that if $k|mn$ and that $\gcd(k, m) = 1$ then $k|n$. However, I noticed an alternative way of proving questions such as: if $2|n$ and $3|n$ show $6|n$ ...
3
votes
1answer
24 views

For $d \in \mathbb{Z}$, if $d\mid a$ and $d\mid b$, show that $d\mid(a+b)$ and $d\mid(a-b)$.

Let $d > 0$ and $d \in \mathbb{Z}$. If $d$ divides $a$ and $d$ divides $b$ then I want to show that $d$ divides $a+b$ and $a-b$. If $d$ divides $a$ then there exist an $m$ such that $a = dm$. If ...
12
votes
3answers
99 views

Prove that $2$, $3$, $1+ \sqrt{-5}$, and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$.

So the Norm for an element $\alpha = a + b\sqrt{-5}$ in $\mathbb{Z}[\sqrt{-5}]$ is defined as $N(\alpha) = a^2 + 5b^2$ and so i argue by contradiction assume there exists $\alpha$ such that $N(\alpha) ...
11
votes
2answers
696 views

What is the remainder when $1! + 2! + 3! +\cdots+ 1000!$ is divided by $12$?

What is the remainder when $$1! + 2! + 3! +\cdots+ 1000!$$ is divided by $12$. I have tried to find the answer using the Binomial Theorem but that doesn't help. How will we do this? Please help.
2
votes
0answers
34 views

divisibility and k-power sum

Let $a_{1},\dots,a_{n},\,n>2$ distinct natural numbers. Prove that if $p_{1},\dots,p_{r}$ are prime numbers and they divide $a_{1}+\dots+a_{n}$ then exists an integer $k>1$ and a prime $p\neq ...
0
votes
1answer
61 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 ...
2
votes
3answers
42 views

Proof of divisibility, given divisibility of a square

The below proof is incorrect. See the answers for more information. This question is in the context of exploring how to explain the process of developing a proof. When reading a proof on the ...
0
votes
0answers
21 views

Check if a set is a groebner basis, based on result of multivariate division algorithm

I've computed that the polynomial $f=x^2y+xy^2+y^2$ when factored with the two polynomials $f_1=y^2-1$ and $f_2=xy-1$, can be rewritten as $f=(xy-1)(x+y)+(y^2-1)(1)+(x+y+1)$ using the multivariate ...
3
votes
5answers
75 views

Show that if $a$ is an integer, then 3 divides $a^3 - a $

Show that if $a$ is an integer, then 3 divides $a^3 - a $ we can write, where $k$ is an integer; $a^3 - a = 3k $ $a(a^2 - 1) = 3k $ Now if $a = k$ then $a^2 -1 = 3$ and $a= \pm2 $ so $ a^3 - a = ...
3
votes
6answers
119 views

Prove that $4^{2n+1}+3^{n+2} : \forall n\in\mathbb{N}$ is a multiple of $13$

How to prove that $\forall n\in\mathbb{N},\exists k\in\mathbb{Z}:4^{2n+1}+3^{n+2}=13\cdot k$ I've tried to do it by induction. For $n=0$ it's trivial. Now for the general case, I decided to throw ...
0
votes
1answer
30 views

Divisiblity of $n$ with $a,b,c$ is relative prime to p

Given an arbitrary prime $p > 2011$. Prove that there exist positive integers $a,b,c$ such that there exists some numbers from $a, b, c$ that are relatively prime to $p$, and for all positive ...
4
votes
2answers
46 views

Prove that $89|2^{44}-1$

Is there any easier (less no. of steps or calculations) proof for this using congruences: $89|2^{44}-1$. My proof: $$2^6\equiv-25\mod89$$ $$2^5\equiv32\mod89$$ Now square both equations: ...
1
vote
5answers
268 views

$24\mid n(n^{2}-1)(3n+2)$ for all $n$ natural problems in the statement.

"Prove that for every $ n $ natural, $24\mid n(n^2-1)(3n+2)$" Resolution: $$24\mid n(n^2-1)(3n+2)$$if$$3\cdot8\mid n(n^2-1)(3n+2)$$since$$n(n^2-1)(3n+2)=(n-1)n(n+1)(3n+2)\Rightarrow3\mid ...
12
votes
7answers
493 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
119 views

Are there infinitely many pairs of primes where one divides one more than the square of the other?

I have the following question on number theory that is eating my head. Are there infinitely many primes $p,q$ such that $p | (q^2 + 1)$ and $q | (p^2 + 1)$? I can see $13,5$ and $2,5$ has the ...
4
votes
0answers
61 views

Reducing multivariate rational fractions to lowest terms

I wish to simplify multivariate rational fractions to a canonical form. Thanks to some very helpful mathematically inclined people who verified that my understanding of Wikipedia was correct, I'm now ...
2
votes
4answers
125 views

Prove that $p$ divides $F_{p-1}+F_{p+1}-1$ [duplicate]

Given the Fibonacci sequence $(F_n)$, defined by $F_0=0,F_1=1, F_{n+2}=F_{n+1}+F_n$, and $p$ an odd prime number, how to prove that $p$ divides $F_{p-1}+F_{p+1}-1$? Is induction a good idea here? ...
4
votes
1answer
294 views

Comparing two definitions of a set of natural numbers

Let $n_1,n_2,N\in \mathbb{N}$. I want to show the following: The two sets \begin{align*} &\Delta(n_1,n_2,N)\\ =& \Big\{ a\cdot b: \quad a\mid {n_1}^2,~a^2 \mid {n_1}^2N ,\gcd\left(N ...
0
votes
1answer
52 views

Highest common factors of polynomials

Let h be a hcf of $f, g \in K[x]$ Then there exists polynomials a and b such that $h = af + bg$ Can anyone explain this theorem to me intuitively?
0
votes
3answers
41 views

Help me answer this Number Theory question on GCD (involves exponents) [duplicate]

Basically I need a good hint how to solve the problem.I have solved it partly. $gcd(2^a-1,2^b-1)=2^{gcd(a,b)}-1$. I have reached till: $gcd(2^a-1,2^b-1)=gcd(2^{a-b}-1,2^b-1)$ How to ...
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
1answer
47 views

To find all positive integers $n$ such that $n|a^2-1 ; a \in \mathbb Z \implies n|a-1 $ , or $n|a+1$

How do we find all positive integers $n$ such that $n|a^2-1 ; a \in \mathbb Z \implies n|a-1 $ , or $n|a+1$ ? I have found that for any odd prime $p$ and $n \in \mathbb Z^+$ , $p^n|a^2-1 ; a \in ...
3
votes
0answers
52 views

Trying to prove a congruence for Stirling numbers of the second kind

I am struggling with a demonstration for this: When $n$ and $m$ are 2 natural integers such that $n-m$ is odd, then the following congruence holds for Stirling number of the second kind ${n \brace ...
2
votes
15answers
804 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 ...