Questions on basic arithmetic, e.g. addition, subtraction, multiplication, division, powers, radicals, etc.

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1answer
42 views

Two conversions to base three yield different results

How are there two different conversion results for the same bases? Am I doing something wrong?
1
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0answers
74 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 ...
4
votes
2answers
109 views

How prove that $ \sqrt[3]{\frac{1}{9}}+\sqrt[3]{-\frac{2}{9}}+\sqrt[3]{\frac{4}{9}}=\sqrt[3]{\sqrt[3]2-1} $

How check that $ \sqrt[3]{\frac{1}{9}}+\sqrt[3]{-\frac{2}{9}}+\sqrt[3]{\frac{4}{9}}=\sqrt[3]{\sqrt[3]2-1} $?
0
votes
1answer
81 views

Factorization of rational powers of rational numbers

If I am not wrong, rational powers of rational numbers can be factorized in an unique way as product of rational powers of different prime numbers: $10^{1/2} = 2^{1/2} \cdot 5^{1/2}$ $(8/9)^{1/6} = ...
0
votes
3answers
85 views

Multiplication of the numbers $1$ to $n$

Let's say I want to find the product of $1,2,3, \dots, 10$. Do I need to do $1 \cdot 2 \cdot 3 \cdot \dots \cdot 10$ manually or is there an easier way to do it? Something like the sumation of $1$ ...
1
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0answers
95 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 ...
6
votes
1answer
121 views

About translating subsets of $\Bbb Z.$

This is a continuation of About translating subsets of R2. Is it possible to find a pair of sets $A,B\subseteq\Bbb Z$ such that A is a union of translated (only translations are allowed) copies of ...
3
votes
4answers
283 views

Diophantine equation abc + abd + acd + bcd= 1

Is there a reference which classifies or at least gives an infinite family of integer solutions to the above equation? A solution to the problem would also be great obviously.
0
votes
2answers
75 views

Simplifying Surd Fractions

can someone show me how to simple surd fractions such as: $$\frac{{8\sqrt 3 }}{2}$$ Can someone please help me here?
0
votes
1answer
56 views

Help with finding the arithmetic mean of all the radii from the center to the edge of an ellipse?

So far I approached this problem computationally, I decided to take all the radii add them up, by distance formula, then divide by the number of radaii. To make the distribution even, I rotated the ...
0
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4answers
58 views

How to work out the percentage?

I'm not exactly sure how to do this. I know the answer is $£64,000$ but whatever I try, nothing is working. i.e. $£80,000$ x $25%$ $= 2,000,000$ devided by $100$ $=$ $20,000$ $£80,000 - £20,000 = ...
1
vote
1answer
63 views

How is this definition of a constant divided by zero called?

I divide a constant by zero. One example is the following: 2/0 My father told me he learned at school earlier that the result is "not defined". If I enter this arithmetic problem in Wolfram Alpha, I ...
2
votes
3answers
170 views

How can this equality be established by elementary algebraic means?

Let $x \geq 1$. Then is it true that $2x^3 - 3x^2 + 2 \geq 1$? If so, how can I show this using only elementary ideas such as factorisation? Of course, I can demonstrate this using the methods of ...
3
votes
1answer
75 views

Euler product of Dirichlet series

Let $f$ be an arithmetic function such $f(n_1n_2)=f(n_1)f(n_2)$ for all $n_1,n_2 \in \mathbb{N}$ with $\gcd(n_1,n_2)=1$. Suppose we know that the Dirichlet series $$F(s) = \sum_{n=1}^{\infty}f ...
-1
votes
1answer
33 views

How to calculate component amounts so their individual additives equal 3%

I have a list of chemical formulas that are each comprised of a number of base components. Two of the base components contain an additive, $X$. This additive needs to exist in each formula at a ...
5
votes
2answers
85 views

How to turn arbitrary fractions into arbitrary egyptian fractions?

I am reading Stillwell's Numbers and Geometry. There is an exercise about Egyptian fractions which is the following: I've tried to do it in the following way - Expressing an arbitrary fraction ...
2
votes
3answers
169 views

How addition and multiplication works

Lets say i am doing 12 + 13 by using the addition method that we know. i mean first we write 13 below 12, then we do 2+3 and then 1+1. The result can be validated as 25 (or true) by doing the counting ...
1
vote
1answer
46 views

Why aren't my % Changes additive?

I'm struggling conceptually with the fact that I have a variable C that is the product of 2 other variables, A and B yet the annual change as a % in C is not the annual change % of A + B. e.g As ...
0
votes
3answers
71 views

When a fraction is raised to a negative exponent, do you normally transform it to 1 over the fraction, or invert the fraction?

My text shows that $$\left(\frac{3a^2}{4b}\right)^{-3}=\frac{1}{\left(\frac{3a^2}{4b}\right)^{3}}.$$ It also shows that $$\frac{1}{\frac{144}{b}}=\frac{b}{144}.$$ In the first equation, it seems ...
1
vote
2answers
93 views

Why does $(3^{1/2})(10^{1/2})=30^{1/2}$ but $(3a^2)(10a^2)=30a^4$?

$(3a^2)(10a^2)=30a^4$? In that equation the exponents are added. Why does $(3^{1/2})(10^{1/2})=30^{1/2}$. In that equation the exponents are not added. Why?
2
votes
2answers
58 views

Square root each term (clarification on polynomials?)

So I'm in Algebra 2, and right now we're learning about conic sections (circles/ellipse/etc). I thought some problems in the workbook looked weird, like this one: ...
0
votes
1answer
37 views

What do you call this arithmetical property and how do you prove it?

I think I can prove this property only when exponents are integers. But here is my example: $a(x-b)^{1/2} = (a^2x-a^2b)^{1/2}$ This type of foiling is weird and I wish to see a proof of this. ...
0
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2answers
56 views

Gambling question: multiply quotes.

Reading various betting forum I came across different threads claiming betting multiple is worse than betting on single events. Could you explain why? [Clairification for the ones not familiar with ...
0
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2answers
108 views

What is the value of $(72^2 - 64^2) : (44^2 - 24^2)$ [closed]

What is the value of $(72^2 - 64^2) : (44^2 - 24^2)$ How to calculate this without calculator?
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votes
3answers
100 views

Learning timetables

This is a basic question, but I am revisiting them due to some examinations I need to take that involves mathematics. I want to be nimble with mental arithmetic so have decided to go back and learn my ...
3
votes
1answer
104 views

An operation with respect to which the set of prime numbers is closed

Like every (semi-)decidable set of natural numbers the set $P$ of prime numbers is diophantine, i.e. there are two polynomials $p(x)$, $q$ with natural coefficients and exponents – the first of ...
6
votes
2answers
209 views

Combination of quadratic and arithmetic series

Problem: Calculate $\dfrac{1^2+2^2+3^2+4^2+\cdots+23333330^2}{1+2+3+4+\cdots+23333330}$. Attempt: I know the denominator is arithmetic series and equals ...
1
vote
1answer
19 views

Find the first $3$ terms of the two possible geometric progressions.

The fourth term of a G.P is $3$ and the sixth term is $147$. Find the first $3$ terms of the two possible geometric progressions. Can you help me find $a$ and $r$? It is too complicated. I took two ...
1
vote
1answer
45 views

Find the sum of the 10th and 11th terms of the G.P.

The third term of a geometric progression of positive terms is $\frac{6}{25}$ and the seventh term is $1\frac{23}{27}$. Find the sum of the 10th and 11th terms of the G.P., giving your answers correct ...
7
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0answers
149 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, ...
1
vote
2answers
50 views

How many wins does my team have to get?

I am watching a tournament. 16 teams are participating in the playoff. Every team play against another once. If you win, you score 1 point, if you lose you dont score any point (the outcome of a match ...
2
votes
3answers
74 views

Square root of a squared number changes sign, which to apply first?

Heres something Ive always found interesting. Supose we have a variable $x$, and $x$ equals a negative number: Say: $$x=-17$$ Now, I can apply a square to both sides of the equation and preserve ...
0
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2answers
44 views

Profit, Loss and percentage problem.

A dishonest businessman professes to sell his articles at cost price but he uses false weight by which he cheats by 10% while buying and 10% while selling. find his profit percentage?
2
votes
1answer
57 views

How many answers can be created using the elementary arithmetic operators?

If I gave you an amount of $n$ numbers, how many anwswer will you be able to create using the elementary arithmetic operators ($+, -, \times, /$)? These are the rules: All numbers ...
2
votes
2answers
68 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
23 views

Square Root in inequalities

I thought I could be able to find an explanation of this (and perhaps I knew it a long time ago) but I cannot find an answer now. If I assume that $b^2 \leq |x|$, then is it true that both $b \leq ...
0
votes
1answer
115 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 ...
2
votes
2answers
44 views

How to calculate price for stamps?

I'm trying to create a programm that has the ability to calculate the most effective way of breaking down the number of stamps (3 and 5 cent) necessary for a consignment. That is - if a transmission ...
0
votes
1answer
37 views

Highest Common Factor to Infinity

Imagine you have a set of integers of x.For example: 7 9 11 13 Let us imagine that y is 1. Then for each nth generation you added 1 to each member of the set, found the HCF of the set and set y to ...
0
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2answers
118 views

Quarters weigh 6 grams while dimes weigh 2 grams.

Quarters weigh $6$ grams while dimes weigh $2$ grams. Tiffany has $\$5.35$ worth of quarters and dimes in her pocket weighing a total of $124$ grams. How many quarters does Tiffany have?
5
votes
2answers
298 views

Prove that $5$ does not divide $52$

If I suppose that $5$ divides $52$, then there would exist an $ s \in \mathbb Z $ such that $ 5s = 52 $. There is no such s, because $5(10) = 50$, and $5(11) = 55$. I'm not convinced with this proof, ...
1
vote
1answer
33 views

Dumb question about the division algorithm.

The theorem about the division algorithm says: Given a, b $ \in \mathbb{Z}, b \neq{0}, $ there exist unique numbers q and r , $q,r \in \mathbb Z $such that $ a = bq + r , 0 \lt r \lt |b| $. Can q ...
5
votes
5answers
324 views

Prove that for all real numbers $x$ and $y$, if $x+y \geq 100$, then $x \geq 50$ or $y \geq 50$.

I'm confused about the following question in my math textbook. Prove that for all real numbers $x$ and $y$, if $x+y \geq 100$, then $x \geq 50$ or $y \geq 50.$ The or is what gets me. For or to be ...
1
vote
1answer
19 views

Commpossed percentage over same base

Maybe this sounds a Little trivial. I have a value and Over this value apply percentage (a), then over this result apply other percentage (b) Could say : Value = 145760 a= 8% b= 40% My actual ...
0
votes
4answers
105 views

How does this sum work?

It seems if base number $a$ is a natural number and the exponent $n$ is an odd number greater than or equal to $3$, then: $f(a, n) =\displaystyle\sum_{i=1}^{a^{n-(n+1)/2}}{(2ai-a)}=a^n$ Such as ...
1
vote
3answers
80 views

SAT Arithmetic Problem [closed]

A total of $k$ passengers went on a bus trip. Each of the n buses that were used to transport the passengers could seat a maximum of $x$ passengers. If one bus had $3$ empty seats and the remaining ...
1
vote
1answer
29 views

Comparison between float point numbers

How can I demostrate that given two numbers A and B, if A <= B, then fl(A) <= fl(B). fl(A) is the float point representation of A. Which is equal to A(1+x), and |x| <= the machine epsilon ...
2
votes
2answers
1k views

How to find a cube root of numbers?

While, I was solving a problem of Chemistry [Solid State] when I encountered an equation like : $$a^3 = 3.612 \times 10^{-23} $$ Where, a is just a quantity [Actually, it is the length of a cubic ...
7
votes
5answers
803 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
2answers
66 views

How do you simplify an algebraic expression?

Please explain how to simplify an expression that is similar to this one $\displaystyle\frac{a+3}{6}+\frac{a-4}{4}+\frac{a+2}{-3}$