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Questions tagged [arithmetic]

Questions on basic arithmetic involving numerical quantities only. Questions involving variable values (other than the result of the operation) should be placed under the (algebra-precalculus) tag.

2
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1answer
48 views

Elliptic curve addition: why does it work in any field?

Supposing I have an elliptic curve E(K): y^2 = x^3 + Ax + B with char(K) != 2,3, why do the formulas for EC addition work in any field. The formulas make sense in ℝ, for example we calculate the slope ...
-1
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0answers
41 views

Prove that $\forall m \in \mathbb{N}^{*},\exists n \in \mathbb{N},\forall k \geq n, p_{k+1}^m<\prod_{i=1}^{k}p_i$

I need to prove : $$\forall m \in \mathbb{N}^{*},\exists n \in \mathbb{N},\forall k \geq n, p_{k+1}^m<\prod_{i=1}^{k}p_i$$ I can prove this assertion using Prime number theorem : For fixed $m$ ...
1
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3answers
33 views

Sum of arithmetic progressions

There is this sum: $$\sum_{i=0}^{n-1}\left(\sum_{j=i+1}^{n-1}(n-j-1)\right)=\frac{1}{6}(n-2)(n-1)n$$ I don't understand how the formula is derived. What I do currently understand is this: For each i,...
1
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1answer
16 views

Does this method always produce the minimal polynomial over $\Bbb{Q}$?

I have used it a few times and so far I have not seen an instance in which it did not give me the minimal polynomial over $\Bbb{Q}$. The method is the following: For an element $a \notin \Bbb{Q}$ set ...
3
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1answer
24 views

Parametrizing the square spiral

Related to this question concerning number spirals I have another one, more specific. While it is rather easy to arrange the natural numbers along an Archimedian spiral by $$x(n) = +\sqrt{n}\cos(2\...
1
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0answers
23 views

Prove that you can multiply using these steps

How to prove this? Think of a simple multiplication problem, I will use $22 \times 7$ Divide the first factor by 2; if the number isn't a whole number, floor it (e. g. $11 / 2 = 5$), until you come ...
0
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1answer
25 views

How to show this rule works for whole numbers to the fifth power?

Today I was shown a rule about natural numbers raised to the fifth power and an interesting method to generate them through the odd numbers. Start with $1 = 1^5$. Then skip the next $T_1 = 1$ odd ...
20
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3answers
482 views

Ways of geometrical multiplication

There are at least five ways to multiply two natural numbers $a$ and $b$ given as integer points $A$ and $B$ on the number line by geometrical means. Two of them include counting, the others are ...
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1answer
51 views

Can somebody elaborate the maths behind this problem?

Theatre Square in the capital city of Berland has a rectangular shape with the size n × m meters. On the occasion of the city's anniversary, a decision was taken to pave the Square with square granite ...
1
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0answers
53 views

Prove that 1+1=2. [duplicate]

Ok... I know that $1+1=2$, but how does that work? What mathematical forces drive this simple, yet profound equation, and how do you prove it? Here is what I did: Let $a=1$. This means that we are ...
1
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3answers
32 views

Is it possible to determine whether a number, can be made from 3 (or 4) others?

I am writing a numbers game involving 3 (or 4) die. The person has to make a number (from 1-25 for 3 die, and 1-100 for 4), and get from one end of a board to the other. For example: Dice 1 = 3, ...
0
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0answers
18 views

Odd and even ordering in subtraction

Creating some sequences of subtractions, and exchanging some numbers of positions, like $ 2 - (3 - 4) \\ 4 - (3 - 2), $ I'm seeing a pattern where substitutions of numbers in odd or even positions ...
5
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3answers
881 views

Largest multiple of $7$ lower than some $78$-digit number?

What I am trying to achieve, is related to cryptography/blockchain/bitcoin . So, the largest number here is huge, in other words: I want to find the largest multiple of 7, which is lower than this ...
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0answers
36 views

If 5 Employees counts 14000 copies in 3 Hrs How much employees required to count 13000 copies in 8 hrs [on hold]

If 5 Employees counts 14000 copies in 3 Hrs How much employees required to count 13000 copies in 8 hrs
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0answers
36 views

Removing specific terms from a product of sums

Consider the following product: $$ \prod_{i=1}^N (a_i + b_i + c_i) $$ Question is: Is there a subtle way to get rid of all terms containing $\ldots a_{i}b_{i+1}\ldots$ $\mathbf{while}$ generating the ...
0
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0answers
8 views

Find price based on margin with percentage removed from price [duplicate]

I am trying to calculate a product's sell price based on the cost and the desired profit margin. Example: Cost \$5 Desired Profit Margin: 50% Sell Price = 5/.5 = 10 Simple enough, however, ...
0
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1answer
20 views

Ambigious Word Questions Involving Ratios (Or Do I Not Know The Conventions?)

I am just curious if I was at fault here for the way I answered these questions, or if the book was at fault for not being clear enough. Perhaps I'm missing something regarding the conventions of ...
0
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0answers
21 views

Why are there 9 functions mapping Set1 = (1,2) to Set2(1,2,3) in the category Set?

Question Answer I dont understand how you can go from a set of two numbers to a set of 3 numbers 9 ways. I count two identity functions, and four other functions mapping 1->2,1->3, 2->1, 2->3. Where ...
5
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2answers
76 views

Associocommutativity

One thing I've noticed is that addition and multiplication both form commutative groups over the reals, but subtraction, division, and exponentiation are neither associative nor commutative. Ignoring ...
0
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1answer
24 views

Computing the present value, compounded monthly

This is a problem from Ross's Elementary Mathematical Finance book: A five-year $\$10,000$ bond with a $10\%$ coupon rate costs $\$10,000$ and pays its holder $\$500$ every six months for five ...
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2answers
46 views

Exercise on polynomial and GCD

Let $f(x), g(x)$ be relatively prime polynomial with coefficients in $\mathbb{Z}$. How can I prove that the GCD $(f(n),g(n)) = O(1)$ for $n \to \infty$, $n \in \mathbb{N}$? Thank you for the help!!
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0answers
22 views

about modular congruence

Consider $p$ a prime and $(1+p\alpha)$ then clearly taking congruences mod $p^m$ $$ \prod_{i=0}^{p^m-1}(1+p\alpha)=(1+p\alpha)^{p^m} \equiv 1+(p\alpha)^{p^m}(\equiv 1 ???) $$ What we can say if we ...
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0answers
30 views

About sums similar to gauss sums

It is well known the case for sums like: $$ \sum_{i=0}^{p^n -1}\zeta^{-ai}, $$ where zeta is a primitive $p^n$-rooth of $1$. But, is there a standard formula for sums like: $$ \sum_{i=0}^{p^n -1}i^...
2
votes
1answer
53 views

Arithmetic Progression Time and Work Problem

A group of men working at the same rate can finish a job in $45$ hours. However, the men report to work, one at a time, at equal intervals over a period of time. Once on the job, each man stays until ...
0
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0answers
29 views

What's with Fractional Subtraction as an action on a number

This a simple (sort of stupid) arithmetic based question that may require just the littlest bit of work. Consider $\cfrac {x}{a}$ I know that division is the number of times I'll have to remove a ...
0
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1answer
23 views

Arithmetic sequences / partial sums [closed]

I am given $a_5=8$ and $a_8=17$ I am asked to calculate the fourth partial sum but have no idea how.
1
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3answers
19 views

Finding a unified ratio from two separate ratios

I'm self-studying with Stroud & Booth's amazing Engineering Mathemathics, 7th edition. I'm stuck at a problem set that gives me two ratios of variables A and B, and B and C respectively, an ...
2
votes
2answers
36 views

Geometric Progression - find ith and r

Hi great if you could please help me solve this: Given: $$a_0=12$$ $$a_5=24$$ $$a_{10} =48$$ thus ratio $$a_0:a_5 = 2$$ $n =$ number of terms in scale $= 5 (a_0, a_1, a_2, a_3, a_4) $ $a_0 =$ ...
2
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2answers
36 views

How does this algorithm calculates the natural logarithm?

This is the pseudocode ...
2
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1answer
37 views

Efficient method to calculate a big n-th root of a number

I need to implement an algorithm to calculate big n-th roots like $\sqrt[\leftroot{-2}\uproot{2}\mbox{13500}]{ 200}$. I have tried the Newton's method but it requires the calculation of very big ...
0
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0answers
17 views

Algorithm for finding numbers that sum up to a certain integer

I am looking to create an algorithm that produces all of the numbers that sum to a number under certain rules. There should be exactly two different natural numbers that are used for the summation. (...
2
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3answers
49 views

Stumped by a pretty basic fraction division

I'm self-studying through Stroud & Booths's amazing "Engineering Mathematics", 7th Edition, and am still on the "Arithmetic" section. Even though I've gone through the whole chapter and a lot of ...
2
votes
1answer
56 views

FOL and Gödel's Incompleteness Theorem 1

My teacher presented Gödel's first incompleteness theorem as follows: (1) First Incompleteness theorem (Gödel [Rosser]): Every [$\omega$-]consistent and reasonably expressive system of arithmetic ...
2
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5answers
640 views

From origin walk halfway to $(8,6)$, turn $90$ degrees left, then walk twice as far.

So, I've spent hours on this question and it's frustrating me way too much so I created an account for StackExchange just to understand how you solve this problem. Let me start by sharing the ...
1
vote
1answer
32 views

why componendo and dividendo is failing in this question

If a positive fraction numerator and denominator is increased by 2 the fraction increases by $\frac{1}{24}$ find the difference between the numerator and denominator, given the sum of Numerator+...
3
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2answers
42 views

Is there a number that has an odd number of tagging zeros, but has a whole square root?

I was doing some math, and I came across the problem $\sqrt{1000}$ And I was thinking about other square roots of numbers with various amounts of zeros. And it occurred to me that it seemed like ...
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0answers
15 views

Coverting denary numbers into octal, binary and hexadecimal form

I'm self studying (after not touching any match for a good 15 years) from Stroud & Booth's amazing "Engineering Mathematics". (This example is from page 55 of the 7th ediction, F.138 of the first ...
0
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5answers
21 views

Arithmetic Sequence problem involving terms of the sequence and the value of that term

The full question is here: In an arithmetic sequence, the first term is $2$ and the second term is $5$. Term number $N$ of the sequence has a value of $M$, such that $M$ is the largest two-digit ...
0
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0answers
24 views

Rate of Percentage Charge

So I was wondering how long it took my iPad to go from 60% to $100\%$ since I got a new USB port. I found it took $216$ minutes, or about $3.6$ hours. My goal was to figure out the time it takes to ...
5
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6answers
84 views

Find the smallest $n \ge 1000$ such the sum $ 1+11+111+⋯+11⋯11 (n$ digits) is divisible by $101$

I get another training problem, for a Romanian 6th grader competition, for which I have no answer. Find the smallest $n \ge 1000$ such the sum $ 1+11+111+⋯+11⋯11 (n digits)$ it is divisible by 101. I ...
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1answer
40 views

There is no $a, b, n \in N, b \ge 2 $ such $a^{b} = 2017^{n} +43$

I found this problem on some training material for a sixth grade Romanian math competition, and I literally have no clue how to approach it. I am not even sure if my interpretation of the problem text ...
0
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1answer
13 views

Find the net rate of interest

A building that costs USD 7,500, rents for USD 62.50 a month. If insurance and repairs amount to $1 \%$ each year, what is the net rate of interest earned on the investment? Method One: I assumed ...
2
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2answers
50 views

If $a, b, c, d$ exists s.t. $p=a^2+kb^2$, $pn=c^2+kd^2$, proof that integer $x, y$ such that $n=x^2+ky^2$ exists.

Question. $p$ is a prime, $k$ is a given natural number. If $a, b, c, d$ exists s.t. $p=a^2+kb^2$, $pn=c^2+kd^2$, proof that integer $x, y$ such that $n=x^2+ky^2$ exists. My approach. Let $n=x^2+ky^2$...
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1answer
307 views
+50

Numbers of distinct values obtained by inserting $+ - \times \div ()$ in $\underbrace{2\quad2 \quad2 \quad2\quad…\quad 2}_{n \text{ times}}$

This question is inspired by How many values of $2^{2^{2^{.^{.^{.^{2}}}}}}$ depending on parenthesis? (By the way, I sincerely hope this kind of questions can receive more attention) Insert $+ - \...
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3answers
25 views

Expanding this completing the square problem.

The following is the final stage of a completed proof, which arrived at a completing the square problem that I'm having trouble figuring out the deriving steps. Could anyone help to show the steps of ...
5
votes
8answers
2k views

How is rearranging $56\times 100\div 8$ into $56\div 8\times 100$ allowed by the commutative property?

So according to the commutative property for multiplication: $a \times b = b \times a$ However this does not hold for division $a \div b \neq b \div a$ Why is it that in the following case: $56 ...
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2answers
45 views

How do Peano's arithmetical axioms guarantee that we can construct the natural number set?

I'm probably not understanding but I don’t see how the axioms can guarantee total construction of the natural number set. The successor function as well as the axiom of induction guarantee that the ...
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2answers
96 views

How to calculate mod of number with big exponent [duplicate]

I want to find $$ 5^{133} \mod 8. $$ I have noticed that $5^n \mod 8 = 5$ when $n$ is uneven and 1 otherwise, which would lead me to say that $5^{133} \mod 8 = 5$ But I don't know how to prove this. ...
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vote
1answer
22 views

Combinatorics leading to fibonacci sequence [duplicate]

I'm trying to generalize the problem that follows oif the number of stairs is "n", the problem is "You climb either 1 or 2 stairs at a time, at any given time. How many ways can you get up 11 stairs?" ...
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2answers
60 views

In a geometric sequence, the sum of the first six terms is 9 times the sum of the first three terms. If the first term is f5, what is the third term?

I solved this question and my answer was 20, but apparently, it's supposed to be -25/2 and I don't understand how that works out. $S_6=9(S_3)$, $t_1=5$ $\frac{1-r^6}{1-r} = 9t_1(\frac{1-r^3}{1-r})$ ...