0
votes
0answers
24 views

What are the patterns in the number of divisors $d(n)$ of the highly composite numbers?

I am trying to understand the patterns in the number of divisors $d(n)$ of the highly composite numbers. The numbers marked with an asterisk are the superior highly composite numbers. The first ...
1
vote
0answers
50 views

Given $m^k\le n <m^{k+1}$ find $x$ and $y$ such that $x\cdot m^k+y=n$

Let $n,m,k\in\mathbb{N}$. Assume $m^k\le n <m^{k+1}$. Find $x,y\in\mathbb{N}$ such that (1) $x\cdot m^k+y=n$ (2) $0<x<m$ (3) $0\le y<m^k$ My question: does there exist a general ...
7
votes
0answers
116 views

Infinite sum involving $q$-adic representations of whole numbers and $q$-factorial numbers

Let $q \in \mathbb{N}_{\geq 2}$. For $n \in \mathbb{N}_0$, let $$\mathrm{fac}_q(n) := \prod_{i=1}^n (1+q+\dots+q^{i-1}) = \prod_{i=1}^n \frac{q^i-1}{q-1}$$ be the $q$-factorial of $n$. In particular, ...
2
votes
2answers
54 views

Geometrical proof of the existence of square roots

This is quite an easy question, but it's been troubling me and I can't manage to work it out. I've been reading the book A Concise Introduction to Pure Mathematics (M. Liebeck), so I'll quote the ...
0
votes
1answer
40 views

How to derive this formula about the bracket function?

Is there a direct way of proving that $$ [nx] = [x] + [x+\frac{1}{2}] + [x+\frac{1}{3}] + \ldots + [x+ \frac{1}{n}]$$ for each real number $x$ and for each positive integer $n$? My effort: Let ...
7
votes
5answers
665 views

Induction hypothesis misunderstanding and the fundamental theorem of arithmetic.

The fundamental theorem of arithmetic is made of two parts: The existence part: There exist primes such that for any natural number $j$, we can write $j$ as a product of prime numbers. The ...
1
vote
1answer
39 views

How to calculate the number of lattice points in the interior and on the boundary of these figures with vertices as lattice points?

We define a point $(x,y)$ in the plane to be a lattice point if both $x$ and $y$ are integers. Now let $$S\colon= \{ (x,y) \ | \ 0 \leq x \leq m, \ 0 \leq y \leq \frac{nx}{m} \}, $$ where $m$ and ...
1
vote
1answer
27 views

“set” addition and divisibility

I have an ordered set with ten natural numbers in it such that when you sum up all ten numbers in the set, it is divisible by 7. Also, if you take the sum of the seven largest numbers in that set and ...
1
vote
2answers
61 views

How to establish this inequality: $(1-a)(1-b)(1-c) \geq 8abc$ for $a+b+c=1$?

Let $a$, $b$, and $c$ be positive real numbers such that $a+b+c = 1$. Then how to establish the following inequality? $$ (1-a)(1-b)(1-c) \geq 8abc.$$ My effort: Since $a+b+c =1$, we can write $$ ...
2
votes
1answer
44 views

What is the most elementary proof of these inequalities?

Let $p$ be a non-zero integer, and let $x_1$, $\ldots$, $x_n$ be $n$ positive real numbers. Then we define the $p$-th power mean $M_p$ of these numbers as $$ M_p \colon= (\frac{x_1^p + \ldots + ...
1
vote
2answers
69 views

How to establish this inequality without using induction?

Given the Fibonacci sequence $a_1 = 1$, $a_2 = 2$, $\ldots$, $a_{n+1} = a_n + a_{n-1} $ for $n \geq 2$, how to derive, without using induction, the inequality $$ a_n < (\frac{1+\sqrt{5}}{2})^n $$ ...
0
votes
2answers
75 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 ...
-1
votes
3answers
111 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
1answer
70 views

Prove this simple arithmetic relation

Prove that if $$a \mid b$$ and $$a \mid c$$ then $$a \mid bx+cy$$ for any integers $x$ and $y$. Here's my proof: $$b = ak$$ $$c = am$$ $$bx+cy = akx+amy = a(kx+my)$$ Notice that $kx+my$ is an ...
1
vote
5answers
82 views

How much zeros has the number $1000!$ at the end?

I know that it depends of the factors of five and two. But the number is too long to figure how much factos of five and two there are. Any hints?
2
votes
2answers
150 views

The number $25!$ has exactly 7 trailing zeros, true or false?

I don't know how to determine it... any hints?
1
vote
1answer
48 views

If $p=1\cdot 3 \cdot 5 \cdot 7 \cdot 9 \cdot … \cdot 2011$, then the units digit of $p$ is five

I know there is a $5$ on the sequence, but i don't know how and why his presence leads to the final units digit of the product.
1
vote
0answers
21 views

Determine an arithmetic relation

Let $f$ be an arithmetic function. Let $p$ be a prime number, $\chi(n)=\left(\frac{n}{p}\right)$ be a primitive Dirichlet character modulo p, where here $~\left(\frac{n}{p}\right)$ is the ...
0
votes
2answers
54 views

Does $\frac{8k-1}{4}$ belongs to $\mathbb{Z}$?

Does $\frac{8k-1}{4}$ belongs to $\mathbb{Z}$ for some $k\in \mathbb{Z}$ ? or we can prove that this never belongs to $\mathbb{Z}$ ?
1
vote
2answers
32 views

Divisibility of the sum of a number and its 'mirror'

I came across the following puzzling problem in an elementary algebra textbook: Problem. Prove that the sum of a two-figure number and the number written with the same digits in the reverse order ...
2
votes
3answers
52 views

proof by contradiction that if a and b are positive integars and $ab >100$ then at least one of the integars a and b is greater than 10 [closed]

does anyone know how to proof by contradiction that if $a$ and $b$ are positive integars and $ab >100$ then at least one of the integars $a$ and $b$ is greater than $10$
2
votes
5answers
55 views

For any prime $p>3$ show that 3 divides $2p^2+1$

Does anyone know how to show this preferable without using modular For any prime $p>3$ show that 3 divides $2p^2+1$
6
votes
3answers
81 views

Show that $9\mid a^2$ if given that $6\mid a$

Does this prove I made seem correct to show that if $6$ divides $a$ then $9$ divides $a^2$ If $6\mid a$, then $a = 6k$ (k is some integer). Then $a^2 = 36k^2 = 9(4k^2)$. Which means that $9\mid ...
0
votes
1answer
50 views

Remainder of $1946^{1972} : 26$

Is this correct? $1946^1 = 22 \mod{26}$ $1946^2 = 22^2 = 484 = 16\mod{26}$ $1946^3 = 22^2 * 22 = 16 * 22 = 14 \mod{26}$ $1946^4 = 22^2 * 22^2 = 16^2 = 22 \mod{26}$ And therefore for any integer ...
0
votes
2answers
37 views

Help on this divisibility Problem

Find all positive integers m and n such that: $$ m+n\mid mn+1 $$ we have according to the condition $$ m+n\mid (m+1)(n+1) $$ any ideas??
2
votes
1answer
31 views

How to find the multiplication of $pq \times abc$ such that the result is producing the same digits from the original problem?

For example: $$65 \times 281= 18265$$ $$65 \times 983= 63895$$ $$72 \times 936= 67392$$ $$87 \times 435= 37845$$ In general: the original figures reappear in the results of each of these ...
2
votes
1answer
46 views

consecutive prime power

I'm interesting on consecutive prime power numbers. I see that there is the Mersenne primes and the Fermat Primes that give solutions and $(8,9)$. In Sloane collection it is referred on A006549 and it ...
1
vote
1answer
47 views

The first odd multiple of a number in a given range

As a part of a programming problem I was solving, we were required to find the first offset of a range at which the number is a odd multiple of another number. For e.g: Take the range $100$ to $120$. ...
0
votes
1answer
71 views

An easy question with integer numbers

I have an easy question of arithmetic. Let $a, b, N$ be integer numbers such that $\mathrm{gcd}(a,b,N) = 1$. Is it true that there exists an integer number $x \in \mathbb{Z}$ such that ...
2
votes
2answers
132 views

99 Ninja missions and a single lantern

From http://blog.liveramp.com/2014/01/30/the-case-of-ninety-nine-ninja/: A team of 99 ninja are each sent out on individual missions by their master. Their master tells the ninja that one of them ...
1
vote
3answers
119 views

Greatest common divisor. Help [closed]

Let $n \in \mathbb{N}$. Prove that $$\gcd(2^n+7^n;2^n-7^n)=1$$ $$\gcd(2^n+5^{n+1};2^{n+1}+5^n)=3\text{ or }9$$
1
vote
4answers
149 views

Which is larger :: $y!$ or $x^y$, for numbers $x,y$.

This is a generalization of this question :: Which is larger? $20!$ or $2^{40}$?. No explicit general solution was presented there and I'm just curious :D Thank-you. Edit :: I want a most-general ...
1
vote
1answer
90 views

Solving $[x]+[x]=[2x]$

Solving the equation $[x]+[x]=[2x]$ Since $[x]$ is the greatest integer function. I tried, $\forall x\in\mathbb{N}$, we have $[x]=x$ and $[2x]=2x$ this implies that $[x]+[x]=[2x]$, but if ...
2
votes
2answers
113 views

Sum of the reciprocals of divisors of a perfect number is $2$?

How do I show that the sum of the reciprocals of divisors of a perfect number is $2$? I tried $d_i\mid n$ with $i\in\mathbb{N},\;d_i\leq n$ then ...
3
votes
3answers
111 views

Find $a,b \in \mathbb{Z}^+$ such that : $\frac{a^{2}-2}{ab+2}\in \mathbb{Z}$

$1$. Find $a;b\in \mathbb{Z}^+$ such that : $\frac{a^{2}-2}{ab+2}\in \mathbb{Z}$ $2$. Find $m;n>1$ such that : $2^m+3^n=k^2$ $(k\in \mathbb{Z})$ Problem 1. I thought : ...
11
votes
2answers
159 views

A problem in fractions from a very old arithmetic textbook

Similar in vein to a problem I posted before here, I would be interested if anyone can give me any pointers as to how one might solve this question from the same arithmetic textbook: "Simplify ...
1
vote
2answers
60 views

Recursive definition of recursively defined operations

The recursive definitions of addition, multiplication, and exponentiation usually stop after exponentiation ("${\small+}1$" to be read as "the successor of"): $x \boldsymbol{+} (y\ {\small+}1) := (x ...
2
votes
1answer
280 views

Prove by induction that $a-b|a^n-b^n$ [duplicate]

Given $a,b,n \in \mathbb N$, prove that $a-b|a^n-b^n$. I think about induction. The assertion is obviously true for $n=1$. If I assume that assertive is true for a given $k \in \mathbb N$, i.e.: ...
0
votes
3answers
85 views

Number of digits of $2^{1000}$ [duplicate]

A friend asks me to find the number of digits of $2^{1000}$. I tried to look for a pattern by calculating the first powers of $2$ but I didn't find it. How should I proceed? Thanks.
1
vote
1answer
49 views

Proof $\displaystyle \text{lcm}(x,y)=\frac{|x\cdot y|}{\text{gcd}(x,y)}$ [duplicate]

Prove: $$ \text{lcm}(x,y)=\frac{|x\cdot y|}{\text{gcd}(x,y)}$$ I used many ways to do it, all failed. One of them was to represent $|x\cdot y|$ as a sum of primes then $\text{gcd}(x,y)$ as a sum ...
0
votes
1answer
120 views

Prove if $a$ & $b$ & $d$ $\in \mathbb{N}^*$ and $gcd(a,b)=d$ and $a=d\cdot k$ and $b=d\cdot n$ then $gcd(k,n)=1$.

Let's define $\mathbb{N}^*$ first: $\mathbb{N}^* = \mathbb{N} - \{0\}$ Prove if $a$ & $b$ & $d$ $\in \mathbb{N}^*$ and $gcd(a,b)=d$ and $a=d\cdot k$ and $b=d\cdot n$ then $gcd(k,n)=1$. ...
1
vote
1answer
94 views

Prove that this sum could'nt be an integrer

let $(p,q)\in\mathbb{N}^{2}$ such that $p\wedge q=1$ ; Prove that $\sum_{j=0}^{j=n}\frac{1}{p+jq}\notin \mathbb{N}$
11
votes
4answers
646 views

Proof that $n^3 + 3n^2 + 2n$ is a multiple of $3$.

I'm struggling with this problem: For any natural number $n$, prove that $n^3 + 3n^2 + 2n$ is a multiple of $3$. That $n^3 + 3n^2 + 2n$ is a multiple of $3$ means that: $n^3 + 3n^2 + 2n = 3 ...
2
votes
2answers
76 views

How to find $p$ and $q$ if we have $\operatorname{lcm}(p,q)=b$ and $p+q=a$ where ($a,b \in \mathbb{N}$) and $p>q$.

What is the general formula to find $p$ and $q$ if we have $\def\lcm{\operatorname{lcm}}\lcm (p,q)=b$ or $\gcd(p,q)$ and $p+q=a$ where ($a,b \in \mathbb N$) and $p>q$? Example: $\lcm(p,q)=84$ and ...
0
votes
1answer
27 views

We suppose: $a, b, c \in \mathbb{N}$ and $b$ and $c$ are multiples of $a$ then $b+c$ is a multiple of $a$.

Are those statements true? We suppose: $ a, b, c \in \mathbb{N}$. We suppose: $b$ and $c$ are multiples of $a$ then $b+c$ is a multiple of $a$. We suppose: $b$ and $c$ are multiples of $a$ and $(b ...
1
vote
2answers
289 views

Remainder when $26^{3008} + 3008^{26}$ is divided by $4$

I want to find the Remainder when $26^{3008} + 3008^{26}$ is divided by $4$. What should I do? Even though I've included the tag modular arithmetic I've very limited knowledge about it. How should I ...
0
votes
2answers
69 views

How to find the remainder of $(2010^{1020} + 1020^{2010})$ divided by $3$

What is the remainder when $2010^{1020} + 1020^{2010}$ is divided by 3?
10
votes
1answer
209 views

Divisibility question from a very old textbook.

I have been looking at a rather old-fashioned book "The Tutorial Arithmetic" (1947), and have been amused by some of the questions in the "Harder Problems" section at the back of the book (some are ...
0
votes
4answers
114 views

If $n$ is coprime with $a$ and $b$, then $n$ is coprime with $ab$

My attempt: Let the prime factorizations of $a,b$ and $n$ be: $$ a = \prod_{i=1}^{k} p_i^{\alpha_i}, \; \; b= \prod_{i=1}^{k} p_i^{\beta_i}, \; \; n=\prod_{i=1}^{k} p_i^{\gamma_i} $$ We know ...
1
vote
2answers
128 views

Is there a compelling reason for the $lcd$ per se and $lcd\equiv lcm$ in fraction arithmetics?

I haven't done arithmetics during the past few years, so I'm filling the gaps before I'm starting out in math in a month, so I have little understanding of the numerics. I've come across such a gap ...