2
votes
1answer
63 views

How do I work out the aspect ratio from the resolution by hand?

For $1024 \times 768$ I can see that $768/1024 = 0.75$, i.e. $\frac34$, so $4:3$ makes sense. How do I do it for other resolutions like $1920 \times 1080$ though?
0
votes
0answers
33 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 $k_i,k_j$. ...
2
votes
2answers
60 views

Proper decimal fraction for $\frac{4n+1}{n(2n-1)}$

Assume I have a function $f(n) = \frac{4n+1}{n(2n-1)}$ with $n \in \mathbb{N} \setminus \left\{ 0 \right\}$. The objective is to find all $n$ for which $f(n)$ has a proper decimal fraction. I know ...
3
votes
3answers
39 views

Divisibility in base $7$ problem

Find all integers between $0 \leq a \leq 2400$ such they are divisible by $8$ and that their base 7 development has at least $3$ equal digits.
2
votes
2answers
57 views

Using GCD/GCF to find number of intersections in a grid

The question I was trying to solve was: A rectangular floor $24×40$ is covered by squares of sides $1$. A chalk line is drawn from one corner to the diagonally opposite corner. How many tiles have ...
4
votes
2answers
98 views

Prove that $3^{n+1}+3^n+3^{n-1}$ is divisible by $13$.

Prove that $3^{n+1}+3^n+3^{n-1}$ is divisible by $13$ for all positive integral values of $n$. I tried: $3^n \cdot 3^1+3^n+3^n\cdot\frac{1}{3}$ Then what should I do next? Help please?
2
votes
2answers
49 views

True or false division algorithm problem

Let a,b,c be integers with a not equal to 0 and (b,c)=1. If a divides the product of bc, then a must divide b or a must divide c. My thoughts: I can prove this if (a,b)=1. but I believe it is false ...
0
votes
2answers
33 views

If u|s and v|t and gcd(s,t) = 1 then gcd(u,v) = 1

Proposition 1. If $\def\divides\mathrel{|}u \divides s$ and $v \divides t$ and $\gcd(s,t) = 1$ then $\gcd(u,v) = 1$. Solution. Assume $u \divides s$ and $v \divides t$. Since $\gcd(u, v) \divides u$, ...
0
votes
0answers
42 views

Proof that $ k^2<2^k$ [duplicate]

I'm struggling with this problem of proof by induction: For any natural number $k\geq 5$, prove that $k^2<2^k$. I assumed that $k^2<2^k$ I want to show that $(k+1)^2<2^{k+1}$ The ...
2
votes
6answers
124 views

Proof that $n^3-n$ is a multiple of $3$. [duplicate]

I'm struggling with this problem of proof by induction: For any natural number $n$, prove that $n^3-n$ is a multiple of $3$. I assumed that $k^3-k=3r$ I want to ...
7
votes
8answers
268 views

Why is $2x^3 + x$, where $x \in \mathbb{N}$, always divisible by 3?

So, do you guys have any ideas? Sorry if this might seem like dumb question, but I have asked everyone I know and we haven't got a clue.
3
votes
1answer
59 views

Solving $\frac{{2{x^3} - 11x + 6}}{{x - 2}}$ using algebraic juggling

Answer: $\eqalign{ & \frac{{2{x^3} - 11x + 6}}{{x - 2}} = \frac{{2{x^2}(x - 2) + 4{x^2} - 11x + 6}}{{(x - 2)}} \cr & = 2{x^2} + \frac{{4x(x - 2) - 8x + 11x + 6}}{{x - 2}} \cr ...
0
votes
1answer
29 views

working out a percentage from my email open rates

i have the following numbers: recipients: $95$ opens: $39$ bounces: $2$ how would i get the percentage value per open? accoridng to this post: ...
3
votes
2answers
139 views

What is an effective means to make divisibility tests a mathematical 'habit', particularly for algebra?

Divisibility tests are a useful problem-solving technique for particularly dealing with larger numbers (thousands etc) and algebraic problems. However, I have always found that many students will just ...
14
votes
7answers
654 views

$n^5-n$ is divisible by $10$?

I was trying to prove this, and I realized that this is essentially a statement that $n^5$ has the same last digit as $n$, and to prove this it is sufficient to calculate $n^5$ for $0-9$ and see that ...
0
votes
1answer
98 views

Finding Pitch Diameter of sprocket

I am currently following a tutorial on Instructables here. In the instructable to find the pitch diameter of a sprocket they use the formula on the above link. the pitch that is used is 12.70, the ...
2
votes
3answers
168 views

Proving that $\frac{n+1}{2n+3}$ is irreducible

I am trying to solve the following problem: Prove that the following fractions are irreducible for any n (n is a natural number and it cannot be null). $\frac{n}{n+1}$ $\frac{n+1}{2n+3}$ ...
6
votes
3answers
408 views

The positive integer solutions for $2^a+3^b=5^c$

What are the positive integer solutions to the equation $$2^a + 3^b = 5^c$$ Of course $(1,\space 1, \space 1)$ is a solution.
0
votes
1answer
140 views

Divide N items into M groups with as near equal size as possible

Im trying to split (say) N pink, fluffy balls into M groups as evenly as possible. Eg: ...
5
votes
4answers
824 views

If $n = m^3 - m$ for some integer $m$, then $n$ is a multiple of $6$

I am trying to teach myself mathematics (I have no access to a teacher), but I am not getting very far. I am just working through the exercises at the end of the book's chapter, but unfortunately ...
4
votes
3answers
123 views

Divide with remainder $\frac{x^2}{x^2 + x + 2}$

I am having a hard time long dividing: $$\frac{x^2}{x^2 + x + 2}.$$ Could someone please show a step by step way to divide this, as I can only get it down to : $1 + \frac{x^2}{x + 2}$. Thank you ...
0
votes
1answer
81 views

Leaving Cert Math Long Division

Solution to problem Hi, I'm correcting my work for study, and I cant get my head around this sum. I understand where the $x^2 + x − cx$ comes from but then when the 6 appears it loses me.
4
votes
6answers
400 views

Prove that $(n-m) \mid (n^r - m^r)$

In respect to a larger proof I need to prove that $(n-m) \mid (n^r - m^r) $ (where $\mid$ means divides, i.e., $a \mid b$ means that $b$ modulus $a$ = $0$). I have played around with this for a while ...