# Tagged Questions

34 views

### Positive integers of sum and products

Find all pairs of positive integers $m$ and $n$ where $m<n$ such that the sum of $m$ and $n$ added to the product of $m$ and $n$ is equal to $2014$ I just thought about this question and ...
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### Pair of positive integers in product sums

I am still not sure on this answer. I would like someone to help me see the solution to his question. I was working on it for a while and it is the only question that I looked at that I can not ...
20 views

### Rational solutions to a system of equations

I have a system of equations \begin{align} xy + 3zw = 0; \\ xz + 2yw = 0; \\ xw + yz = 0. \\ \end{align} Plugging it into a CAS, I see that all the rational solutions to this system have ...
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### Show that ${n \choose 1} + {n \choose 3} +\cdots = {n \choose 0} + {n \choose 2}+\cdots$ [duplicate]

Show $${n \choose 1} + {n \choose 3} +\cdots = {n \choose 0} + {n \choose 2}+\cdots$$ A hint is given to consider the expansion $(x-y)^n$ However, when I plug in a number for $n$, I don't get an ...
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### Determine the number of digits in $4^n$

Let $n$ be a natural number. How can we determine the number of digits in $4^n$? For example $4^{20}$ has $13$ digits.
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### Why is it impossible to find natural numbers $a$ and $b$ such that $12b^2=a^2$?

This was a question in the exercises for an EdX course by Prof Starbird on Effective Thinking through Mathematics which was long over, but I am working through the course at my own pace. I feel that ...
55 views

### decomposition into three squares

Doing a coding assignment. And it's basically having a user enter $n$. Then I need to provide (If it exists) $$n = x^2 + y^2 + z^2.$$ Not really sure how to approach this. Any ideas?
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### What is the meaning of this Wolfram Alpha result when calculating $3^p = 4^q$?

I would like to know are the some $p \in \mathbb{N}$ and $q \in\mathbb{N}$ for $3^p = 4^q$ except the trivial $p = q = 0$. So, I entered the expression into Wolfram Alpha, which returned the result ...
66 views

### Simplifying radical expressions such as $\sqrt{80}$

I am having trouble simplifying a radical expression, such as say...$\sqrt{80}$. What I do is firstly, I do 80/2, then 80/3, then 80/4, then 80/5...etc until I find the largest number that can be ...
186 views

### The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer

Prove by induction that this number is an integer: $$u_n=(3+\sqrt{5})^n+(3-\sqrt{5})^n$$ Progress I assumed that it holds for $n$ and I tried to do it for $n+1$ but the algebra gets quite messy and ...
100 views

### How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?

I am reading a chapter about mathematical proofs. As an example there is: Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for ...
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### Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which says what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$? a. 1 b. 3 c. 2 d. empty I know that by considering certain cases, for example when $x<0$ or ...
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### how shall i find the $n$-th term of this,

How shall I find the $n$-th term of this: $\sqrt{1+2}$ $\sqrt[3]{1+2+3}$ $\sqrt[4]{1+2+3+4}$ $\sqrt[5]{1+2+3+4+5}$ $\sqrt[6]{1+2+3+4+5+6}$ $\sqrt[7]{1+2+3+4+5+6+7}$ all the way to ...
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### Find integral solutions for $2x^2+y^2=2\times(1007)^2+1$

Find integral solutions to the equation $$2x^2+y^2=2\times(1007)^2+1$$ I tried: I rewrote the equation as $2x^2+y^2=2028099$. I found that $y_{max}=1424$ and $y$ must be odd, so I set ...
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### Is this real number an integer?

Is this real number : $$\Big(2+\frac{10}{9}\sqrt{3}\Big)^{1/3}+\Big(2-\frac{10}{9}\sqrt{3}\Big)^{1/3}$$ an integer ? I've tried different factorization, but nothing seems to work.
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### Mathematical way to solve integer numbers $217 = (20x+3)r+x$

Is there any mathematical way to find the integer numbers that solve the following equation: $$217 = (20x+3)r+x$$
56 views

### How to simplify the formula for $n$th Fibonacci number when $n=2$?

When n is equal to 2 how do I simplify when the $n=2$ is put into the equation below (by the way I have to prove this formula by induction that when n= any number it will equal that number) ...
84 views

### searching smallest number that has $40$ distinct positive divisors

What is the smallest natural number such that it has $40$ distinct positive (integer) divisors (inclusive of $1$ and itself? At first I was stunned of seeing the problem.It's not possible to find ...
### Given $n$, find smallest number $m$ such that $m!$ ends with $n$ zeros
I got this question as a programming exercise. I first thought it was rather trivial, and that $m = 5n$ because the number of trailing zeroes are given by the number of factors of 5 in $m!$ (and ...