3
votes
5answers
560 views

The sum of three consecutive cubes numbers produces 9 multiple

I want to prove that $n^3 + (n+1)^3 + (n+2)^3$ is always a $9$ multiple I used induction by the way. I reach this equation: $(n+1)^3 + (n+2)^3 + (n+3)^3$ But is a lot of time to calculate each ...
0
votes
1answer
21 views

Position of switches based on divisibility

There is a set of $1000$ switches. Each has four different positions, called $A$, $B$, $C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to ...
0
votes
2answers
62 views

Find all values of for which the ratio is an integer

Find all values of $n$ for which, $$\dfrac{(\dfrac{n+3}{2}) \cdots n}{(\dfrac{n-1}{2})!}$$ is an integer. I have tried the problem for some primes. Each time it seemed true. But I still ...
1
vote
2answers
31 views

GCD question involving coprimes

How can I prove that if $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \times \gcd(b, c)$? By eea there exists $ax+by=1$ from $\gcd(a,b)=1$ so a and be are co-primes there also exists $dk=a$ and ...
0
votes
1answer
16 views

Proving $k$ is a common divisor of $a$ and $b$ iff $k\,|\gcd(a,b)$ — converse

I'm currently trying to solve the converse of this statement is true after proving the normal version is true. If $k$ is a common divisor of $a$ and $b$, then $k \,| \gcd(a, b)$ So far I know the ...
2
votes
1answer
17 views

Questions relating to gcd

Assume a, b and c are positive integers. 1) Suppose that a | b. Show that gcd(a, c) ≤ gcd(b, c). 2) Suppose that a ≤ b. Is it necessarily true that gcd(a, c) ≤ gcd(b, c)? I'm having trouble with ...
3
votes
1answer
60 views

If $k$ is an odd number then $3k^2 +16$ is not a perfect cube

I am pretty sure that the title is true. Could anybody please prove it? I am particularly interested in a proof that mostrly relies on divisibility.
1
vote
1answer
41 views

Prove that $f(n,p)$ is a non-square integer

Let, $$f(n,p)=(n+1)(n+2) \cdots (n+p-1)$$ Then show that $f(n,p)$ is a not a perfect square for all $n \in \mathbb{N}$ and for all odd primes $p$. Consider only the cases when ...
2
votes
5answers
61 views

Proof regarding divisibility

if $n\in\mathbb{Z}$, then $4$ does not divide $(n^2 - 3)$ I'm not sure how to approach this question, I know how to do questions that involve proving that it does divide but I'm unsure of how to do ...
1
vote
2answers
30 views

Contrapositive proof using rule of divisibility

Suppose $x,y,z$ are integers and $x \neq 0 $ if $x$ does not divide $yz$ then $x$ does not divide $y$ and $x$ does not divide $z$. So far I have: Suppose it is false that $x$ does not divide $y$ and ...
0
votes
2answers
59 views

Prove or disprove this implication

Prove or disprove: If $x, a, b > 0$ are integers such that $$\gcd(x-a, x+b) = 1\ \ \mbox{and}\ \ \gcd(2x-a, x+b) > 1,$$ then $$a+b = x.$$
2
votes
1answer
44 views

$[n,n+1]=\text{ ??????}$

I know the answer is $n(n+1)$, but I'm having trouble formulating an argument. I know by the definition, if I let $h=[n,n+1]$ $$h=nk_1, h=(n+1)k_2$$ $$nk_1=(n+1)k_2$$ ...
0
votes
1answer
34 views

What are the smallest numbers $n$ such that $\dfrac{d(n)}{\ln(n)} \geq k$ where $d(n) = \sigma_0(n)$ is the number-of-divisors function?

I have calculated $\dfrac{d(n)}{\ln(n)}$ on a few highly composite numbers up to 5040. Here is what I got: $\dfrac{d(120)}{\ln(120)} = 3.3420423$ $\dfrac{d(360)}{\ln(360)} = 4.0773999$ ...
2
votes
2answers
29 views

highest power of prime $p$ dividing $\binom{m+n}{n}$

How to prove the theorem stated here. Theorem. (Kummer, 1854) The highest power of $p$ that divides the binomial coefficient $\binom{m+n}{n}$ is equal to the number of "carries" when adding $m$ ...
0
votes
1answer
17 views

$B$-powersmooth number divides $\mathrm{lcm}(1,2,3,\ldots B)$

Let $M$ be $B$-powersmooth (ie. all prime powers in $M$'s factorization are $\le B$). I want to prove that $M \mid \mathrm{lcm}(1,2,3,\ldots, B)$. I thought it would be easy to prove this using ...
1
vote
1answer
39 views

$nc_i\mid\prod_{i=1}^3(nc_i+1)-1$ iff $\exists c\in 6\mathbb{N}:c_i=ic$

Let $c_1<c_2<c_3$ be natural numbers and $$C_n=\prod_{i=1}^3(nc_i+1)-1\;\;\;\;\;(n\in\mathbb{N})$$ I want to show that it holds $$\forall n\forall i : nc_i\mid ...
0
votes
2answers
69 views

Prove that if $n^2 - 1$ is divisible by a prime number $p$ such that $n - 1$ is not divisible by $p$, then $n + 1$ is also divisible by $p$.

If this proposition is false, please give at least $3$ counter-examples, and try to modify the proposition so that it becomes true. If the proposition is true, please try to prove this even more ...
2
votes
3answers
155 views

Find positive integers $(x,n)$ such that $x^{n} + 2^{n} + 1$ is a divisor of $x^{n+1} +2^{n+1} +1$

Find all positive integers $(x,n)$ such that $x^{n} + 2^{n} + 1$ is a divisor of $x^{n+1} +2^{n+1} +1$ I encountered this question in one of my monthly assignments. Unfortunately, I don't know ...
4
votes
1answer
117 views

$\sum\limits_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}$ isn't divisible by 5

I have no idea Prove that for any $n$ natural number this sum $$\sum\limits_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}$$ isn't divisible by $5$. $\begin{array}{l} \left( {1 + x} \right)^{2n + 1} - ...
0
votes
1answer
68 views

Why does the number of divisors of a superior highly composite number is always a highly composite number up to 720720 ? (the only exception is 120)

I've calculated the number of divisors of every superior highly composite number up to $10^{27}$: http://oi59.tinypic.com/ndaijo.jpg The number of divisors of a superior highly composite number is ...
0
votes
0answers
73 views

Why isn't $1$ a superior highly composite number?

A superior highly composite number is a positive integer $n$ for which there is an $\epsilon>0$ such that $\dfrac{d(n)}{n^\epsilon} \geq \dfrac{d(k)}{k^\epsilon}$ for all $k>1$, where the ...
0
votes
1answer
75 views

Have you seen this theorem before? (GCD divides, neccessary & sufficient condition)

Conjecture. Let $a,b, c\in \Bbb{Z}, b \neq 0$, The following conditions are equivalent: (1) $d = \gcd(a,b)$ divides c. (2) There's a polynomial in $f \in \Bbb{Z}[X,Y]$ with $c$ constant term, such ...
-1
votes
3answers
108 views

Prove that if $n+1$ is divisible by $m$ then $n^2-1$ is also divisible by $m$. [closed]

Prove that if $n+1$ is divisible by $m$ then $n^2-1$ is also divisible by $m$. And prove that if $n^2-1$ is divisible by $m$ then $n+1$ is also divisible by $m$.
2
votes
3answers
77 views

If $3$ divides $q^3$, is it true that $3$ divides $q$?

I think this is true because of prime factorisations, i.e. If $3$ a factor of the prime factorisation of $q^3$, then $3$ is a factor of the prime factorisation of $q$. Therefore If $3$ divides ...
1
vote
2answers
65 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 ...
-11
votes
1answer
115 views

How many divisors do these big superior highly composite numbers have? [closed]

448806242393308800 18401055938125660800 791245405339403414400 37188534050951960476800 185942670254759802384000 9854961523502269526352000 581442729886633902054768000 1162885459773267804109536000 ...
0
votes
0answers
65 views

Among the superior highly composite numbers, which are the most divisor dense numbers?

I’m searching for the most divisor dense natural numbers. Firstly we have the highly composite numbers: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, … But ...
0
votes
1answer
27 views

For how many values of $a,b,c\in(1,2\ldots,p-1)$ does $p$ | $({a^2}-bc)$ where $p$ is an odd prime number

In a mock test for an entrance exam I am preparing for came the following question: Let $p$ be an odd prime number and $T_p$ be the following set of matrices $$ T_p= \left( ...
0
votes
1answer
43 views

Verify my proof on elementary number theory

I've tried to prove this theorem, which is very simple, but is a kind of practice for me. Let $a,b$ be two positive integers. Therefore, if $a+b$ is a composite number, $frac(\frac{a}{l}) + ...
0
votes
0answers
49 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 ...
0
votes
0answers
32 views

The elegant expression in terms of gcd and lcm - algebra - (2)

Definition: suppose a quantity $P$ is identified by $$ \frac{P}{k}\simeq \frac{P}{k}+1 $$ what we mean is that $$ P= 0\pmod{k}. $$ That means that when $P \to P+k$, then $$ \frac{P}{k}\to ...
2
votes
1answer
39 views

The elegant expression in terms of gcd and lcm - algebra

Given three positive integer numbers $k_1$, $k_2$, $k_3$, we may denote their greatest common divisor(gcd) by $\gcd(k_i,k_j)\equiv k_{ij}$ for gcd of a two pair of number $k_i,k_j$. ...
0
votes
1answer
30 views

Number Theory Divisibility Question

(From Math Challenge II Number Theory packet) Given that $a,b,n$ are positive integers. Assume that for any positive integer $k\neq b, (k-b)\mid(k^n-a)$, the which of the following must be true? ...
1
vote
2answers
63 views

How to prove that at least one of $a,b,c,d$ is not divisible by $ad-bc$ if $ad-bc>1$?

we have $ad-bc >1$ is it true that at least one of $a,b,c,d$ is not divisible by $ad-bc$ ? Thanks in advance. Example: $a=2$ , $b = 1$, $c = 2$, $d = 2$, $ad-bc = 2$ so $b$ is not divisible by ...
0
votes
2answers
54 views

Divisibility of sum of powers: $\ 323\mid 20^n+16^n-3^n-1\ $ for which $n?$

I found this question in my Math Challenge II Number Theory packet: Find all positive integers $n$ that satisfy $323|20^n+16^n-3^n-1$. I don't even have any idea how to approach this question. Any ...
0
votes
2answers
36 views

Remainder question with $6!$ and 7

Find the remainder when $6!$ is divided by 7. I know that you can answer this question by computing $6! = 720$ and then using short division, but is there a way to find the remainder without using ...
0
votes
1answer
65 views

Pythagorean quadruple generators with a gcd relation

For non-negative integers $m,n,q,p$ with $\gcd(m,n,q,p)=1$, assume we have: $$\gcd(mq+np,b)=|nq-mp|$$ for some integer $$b<mq+np$$ and that $$8\nmid\,mq+np,$$ $$m+n+p+q\equiv 1\mod 2.$$ Can ...
1
vote
3answers
37 views

Prove that if m is prime and m|kl then either m|k or m|l.

Proofs homework question, here's what I've figured out thus far. Suppose m doesn't divide k. We need to then prove that m|l. If m doesn't divide k and m is a prime then we know m and k are co-prime ...
0
votes
4answers
53 views

How to prove that if a number is divisible by two other numbers, then it is divisible by there product

I would like to prove if $a \mid n$ and $b \mid n$ then $a \cdot b \mid n$ for $\forall n \ge a \cdot b$ where $a, b, n \in \mathbb{Z}$ I'm stuck. $n = a \cdot k_1$ $n = b \cdot k_2$ $\therefore a ...
1
vote
2answers
100 views

A puzzle on elementary number theory

I have been stuck with the following puzzle for some time. I could not prove it, nor could I find a counter example. I would be grateful to get some help on this.
1
vote
2answers
56 views

Solving a problem with a diophantine equation without trial and error.

I have the following problem: A teacher bought toys for the students of an academy, every toy for a boys costs $290$ and every toy for a girl costs $330$. If he spends $24300$, how many of each ...
0
votes
3answers
23 views

Let $a$ be a positive integer. The sum of $a$ consecutive integers is divisible by $a$ if and only if $a$ is odd.

How would one prove this? Other than using cases to prove the if and only if part, how would I prove each case to complete the proof?
0
votes
3answers
108 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
2answers
54 views

Divisibility exercise

Now, I have an idea how to attempt this question with modulo arithmetic, but I was thinking if there was a solution that did not involve modular arithmetic. If $7 |(b^2+c^2)$ iff $7|b$ and $7|c$. I ...
1
vote
1answer
68 views

Prove something is divisible by a prime

Let $p$ be a prime. Prove that $\sum_{k=j}^{p-1} \frac{k!}{ j! (k-j)! }$ is divisible by $p$ $\forall$ $j \in \{0, ..., p-2\}$. Where this problem comes from: I am trying to prove that ...
1
vote
5answers
87 views

prove that $3$ does not divide $n^2+1$

How do I prove that $3$ does not divide $n^2+1$, for all $n\in\mathbb{Z}$, thought of in separate cases, but did not get, induction also was unable to ....
2
votes
1answer
82 views

Prove $\gcd(ka,kb) = k*\gcd(a,b)$

For all $k > 0,\ k\in \Bbb Z$ . Prove $$\gcd(k*a,\ k*b) = k *\gcd(a,\ b)$$ I think I understand what this wants but I can't figure out how to set up a formal proof. These are the guidelines we ...
1
vote
1answer
86 views

Why does Euclid's algorithm taken one step past the GCD seem to yield the LCM?

In class we are going over Euclid's Algorithm. For example, we learned that for integers $m$, $n$:$$\gcd(m,n) = sm + tn$$ Where $s$ and $t$ are integers that can be plugged in to satisfy the ...
0
votes
2answers
46 views

Let n be an integer greater than 3. Find a formula for gcd(n, n + 3)

Let $n$ be an integer greater than $3$. Find a formula for $\gcd(n, n + 3)$ for each of the cases : $1)$ $n \equiv 0\mod 3$ $2)$ $n \equiv 1\mod 3$ $3)$ $n \equiv 2\mod 3$ Any help would be greatly ...
0
votes
2answers
48 views

Feedback on Euclidean Algorithm: $gcd(277, 301)$

Ans: $301 =277 \cdot 1 + 24$ $277 =24 \cdot 11 + 13$ $24 = 13 \cdot 1 + 11$ $13 = 11 \cdot 1 + 2$ $11 = 2 \cdot 5 + 1$ $2 = 1 \cdot 2 + 0$ Is this correct?