Integer Definition and 620 Threads

Can the No :$4\sqrt{4-2\sqrt {3}}+\sqrt{97-56\sqrt 3}$ be an iteger ,if yes prove it if no then prove it again

For a pair of positive integers $(a,\,b)$, $Q(a,\,b)$ is defined by $Q(a,\,b)=\dfrac{a^2b+2ab^2-5}{ab+1}$. Let $(a_1,\,b_1),\,(a_2,\,b_2),\,\cdots, (a_n,\,b_n)$ be all pairs of positive integers such that $Q(a,\,b)$ is an integer. Calculate $\displaystyle \sum_{i=1}^n a_i$,

Not a question as such, but an interesting notion that I came upon (maybe some other people would find it interesting too). It seems to have been introduced in 1950's and seems a good amount of work has been done on it. For example: 12=(1+1+1)*(1+1+1+1) So the complexity of 12 is 7 since it can...

Prove that for any arbitrary odd x, that x^2 is also odd. By definition an odd number is an integer that can be written in the form of 2k + 1 for some integer k. This means that x = 2k + 1 where k is an integer So let x^2 = (2k + 1)^2 we then get 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1, This is where...

Find all the integer values of $m$ for which the equation $\left\lfloor \dfrac{m^2x-13}{1999}\right\rfloor=\dfrac{x-12}{2000}$ has 1999 distinct real solutions.

Find all integer solutions to the equation $4x+y+4\sqrt{xy}-28\sqrt{x}-14\sqrt{y}+48=0$.

Let $a,\,b,\,c$ be three distinct integers and $P$ be a polynomial with integer coefficients. Show that in this case the conditions $P(a)=b,\,P(b)=c,\,P(c)=a$ cannot be satisfied simultaneously.

Proof: Let a be a even positive integer of the form a=2m & b of the form b=2n (This is where b is a even positive integer) ab = 2m*2n = 2(mn) = Let k = mn = 2k Therefore, ab is even. Let a be a even positive integer a=2m & b be a odd positive integer b = 2n+1 ab = (2m)*(2n+1)...

I came across a rather strange thing in an introductory class I still don't understand. There was a statement that $$lim_n (2+ \sqrt(2))^n $$ is an integer. I recalled that I never understood this and just recently tried to take the limit but just get that the expression diverge? Which I think...

Find all integer solutions of the system of equations $x+y+z=3$ and $x^3+y^3+z^3=3$.

I had learned how to find this out in the past, but forgot now. Precisely, I'm trying to find in how many different ways I can express the number 24 as a sum of two integers ranging from 1 to 24. For example, 24 = 24 = 23 + 1 = 12 + 12...

Hey! 😊 Question 1: We consider $\frac{2n-1}{n+7}$. For which $n$ is this term an integer? I have done the following: We set $n+7=m \Rightarrow n=m-7$. Then we get $$\frac{2n-1}{n+7}=\frac{2(m-7)-1}{(m-7)+7}=\frac{2m-15}{m}$$ So $m$ has to be a divisor of $15$, i.e. $m\in \{1,3,5,15\}$...

Find the largest even integer which cannot be written as the sum of two odd composite numbers.

Find all positive integers $a$ and $b$ such that $\dfrac{a^2+b}{b^2-a}$ and $\dfrac{b^2+a}{a^2-b}$ are both integers.

I have found code to find simply the minimum numbers needed, but I need to do it without repetition given the nature of an electric circuit. I hope that is a sufficient enough explanation of the problem. Despite being an engineering project this aspect is more mathematical.

Find all integer solutions (if any) for the equation $(A+3B)(5B+7C)(9C+11A)=1357911$.

Two integers will be taken from 1 to 50, where at least one of them should be a square number and sum of them should also be a square number. How many different pair like this can be found? Will I count (9,16) and (16,9) as one ?

Hi all. I made a program of DFT, so I made the power spectrum of a sin wave. This is the sin wave I used. All data number ##N=100## and the frequency of sine wave is 4.5Hz. And the power spectrum is this. The wave number is not integer so the spectrum has the side lobe. But I think this is...

Hello everyone, I want to calculate the following limits: \[\lim_{x\rightarrow \infty }\frac{[x\cdot a]}{x}\] using the sandwich rule, where [xa] is the integer part function defined here: Integer Part -- from Wolfram MathWorld I am not sure how to approach this. Any assistance will be...

find me an integer that isn't divisible by 1.

Hello, In grade 11 of high school, I encountered this linear programming problem on my textbook: The "alternative solution" described in the textbook as follows: Let: - ##x## : amount of plant A - ##y## : amount of plant S - ##L## : garden area - ##L_x## : area of garden for one plant A -...

im trying to complete mips program code about a calculator program that can calculate integer addition / subtraction written using the MIPS assembler. im having hard times to debug this. The input is given to the array of Formula char (base address $ s0) in the form of a formula. The null...

Let Z = set of real numbers Determine if (27/4)/(6.75) is a whole number, natural number, integer, rational or irrational. I will divide as step 1. 27/4 = 6.75 So, 6.75 divided by 6.75 = 1. Step 2, define 1. The number 1 is whole or natural. It is also an integer and definitely a rational...

x^2 - x -3 + 2c = 2x(ax+b) x^2 -2ax^2 - 2bx - x - 3 + 2c = 0 x^2(1-2a) -x(1+2b) -3 + 2c =0 Using girard r1+r2 = (1+ 2b)/(1-2a) r1xr2 = (-3 +2c)/(1-2a) After this I am stuck. Thank you.

$n$ is a positive integer with the following property: If the last three digits of $n$ are removed, $\sqrt[3]{n}$ remains. Find with proof $n$. Source: Nordic Math. Contest

Homework Statement [/B] So, this is a question I have in one of my assignments and I'm really going round in circles with it. Any pointers or links to additional reading would be gratefully received! This is a pass criteria question on a General Engineering HNC. Homework Equations [/B] Why do...

Prove the existence of a positive integer divisible by $2019$ the sum of whose decimal digits is $2019$.Source: Nordic Math. Contest

Dear Everybody, I don't know where to begin. So Here is the problem: $\newcommand{\Z}{\mathbb{Z}}$ Prove that if $[a]$ and $[b]$ are in ${\Z / n\Z}^{\times}$, then $[a] \times [b]$ is in ${\Z / n\Z}^{\times}$. Thanks, Cbarker1

I identified what appears to be a partitioning of all integers > 1 into mutually disjoint sets. Each set consists of an infinite series of integers that are all the powers of what I am calling a "root" r (r is an integer that has no integer roots of its own, meaning: there is no number x^n that...

Homework Statement Let ##a,b## be squarefree integers and set ##R = \mathbb{Z}[\sqrt{a}]## and ##S = \mathbb{Z}[\sqrt{b}]##. Prove that a) There is an isomorphism of abelian groups ##(R,+) \cong (S,+)##. b) There is an isomorphism of rings ##R\cong S## if and only if ##a=b##. Homework...

NOTE:This is not a homework question! This is just a topic that I like very much,but don’t have the programming ability to do many of them.That’s why I post this thread. C++ is a language without built-in big integer calculation functions,so building ones that can do such job is a great way to...

Why ln(k) when k is a possitive integer, ln(k) is a complex number?

Is there a way to find the integer of a real number? Of course without using the [x] function. What I am looking for here is an algebraic formula.

Homework Statement I've tried hours and hours to model this problem but i didn't succeed. Can you help me ? We want to realize n projects in the next T years. For each project, a profitability index pi is known, which expresses the expected final gain (in Euro) and a cost profile ai =...

Homework Statement # of integer solutions of x1+x2+x3+x4 = 32 where x1,x2,x3>0 and 0<x4≤25 Homework EquationsThe Attempt at a Solution So in the case where x4 = 25 we have x1+x2+x3= 4 in the case where x4 = 24 we have x1+x2+x3 = 5 ... in the case where x4 = 1 we have x1+x2+x3 = 28so...

Suppose a and b are integers that divide the integer c If a and b are relatively prime, show that $ab / c$ Show by example that if a and b are not relatively prime, then ab need not divide c let $$a=3 \quad b=5 \quad c=15$$ then $$\frac{15}{3\cdot 5}=1$$ let $$a=4 \quad b=6 \quad c=15$$ then...

I don't usually need help in locating software, but I'm having a heck of a time tracking down a good open-source bit of software which solves integer programming problems using arbitrary precision! If I don't find one soon, I'll need to write it myself. Which I don't mind, but it's silly to...

Homework Statement I've got to integrate the following $$ \int dx =\int \frac {d\phi} {\phi \sqrt {1 - \phi²}}. $$ Homework Equations [/B] I already know the answer but not how to get it. The answer that I got from solution is ## x = \operatorname {arcsech}{\phi} ##. The Attempt at a...

Homework Statement Consider ##u\left(x\right)=2\left[\frac{-x}{4}\right]## (a) Find the length of the individual line segments of the function, (b) Find the positive vertical separation between line segments. Homework Equations The output of Greatest Integer Functions are always integers. The...

I would like to know if there is an official name for the class of integers that are (not) perfect powers. A perfect power is a number that can be expressed as xn, where x and n are both integers > 1. I have been calling these integers "roots" - since they do not have any integer roots of their...

Here is this week's POTW: ----- If $n$ is a positive integer, evaluate $$\int_{0}^\infty \frac{dx}{1 + x^n}$$ ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...

I am wondering about the integer solutions to ##a^3+b^3+c^3=d^3~##. By trial and error I stumbled upon ##3^3+4^3+5^3=6^3##. I find this equation remarkable in that not only the four integers are consecutive, but also because the three integers on the left form the well known Pythagorean...

I know there are many proofs of this I can google, but I am interested in a particular one my book proposed. Also, by countable, I mean that there is a bijection from A to ℕ (*), since this is the definition my book decided to stick to. The reasoning is as follows: ℤ is countable, and so iz ℤxℤ...

Determine the integer $n$ with the properties: a). $n$ is a prime less than $6000$, b). the number formed by the last two digits of $n$ is $< 10$, and c). if the decimal digits of $n$ are reversed to obtain $N$, then $N − n = 999$.

I have in the back of my head the statement that for every finite sequence of positive integers there exists a pattern (i.e., a generating formula). While this sounds reasonable, I am not sure whether it is true, and if it is true, what the source for this statement is, and how the correct...

Let $a = \sqrt[3]{7+5\sqrt{2}} + \sqrt[3]{7-5\sqrt{2}}$. Show (without the use of a calculator), that $a$ is an integer.

Show that for integer a,b,c the product $abc(a^3-b^3)(b^3-c^3)(c^3-a^3)$ is divisible by 7

Hello! (Wave) We want to find an efficient algorithm that checks whether an odd number is a prime or not. In order to obtain such an algorithm, one tests the congruence $(X+a)^n \equiv X^n+a$ not "absolutely" in $\mathbb{Z}_n[X]$, but modulo a polynomial $X^r-1$, where $r$ have to be chosen in...

Homework Statement Formulate as a mixed integer programming problem but do not solve. Maximize ##x_1 + x_2## subject to ##2x_1 + 3x_2 \le 12## or {##3x_1 + 4x_2 \le 24## and ##-x_1 + x_2 \ge 1##} ##x_1, x_2 \ge 0## Homework EquationsThe Attempt at a Solution if the first constraint is met, we...