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

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2
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71 views

Solve $x + y + z = xyz$ such that $x , y , z \neq0$

I came across the equation $x+y+z=xyz$ such that $x , y , z \neq 0$. I set $x=1, y=2, z=3$ but how can i reach formal mathematical solution without " guessing " the answer ? Thank you
0
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1answer
20 views

modular problem in arithmetic

hello can someone please help me to solve this problem: 2008 mod 71, 9 square mod 41, 34 suare mod 71 b)determine all a and b that verify a square mod 41=40 b square mod 71=20 ...
0
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1answer
17 views

a tax Deferred keogh account

Suppose you contribute $20,000 in an account at the end of the year.How much would you have at the end of 20 years if the account pays 8% annual interest.
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3answers
49 views

How do I compute $-6(-4)^{n-1} + 8(-4)^{n-2}$?

How do I compute $-6(-4)^{n-1}$ + $8(-4)^{n-2}$ ? I recall that as long as the number from both operands (in this case: -4) are the same, I can actually "add" them together. But the problem is the -6 ...
1
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4answers
240 views

Why is modulus not a basic arithmetic operation?

In school I learned that there are four basic arithmetic operations: addition, subtraction, multiplication, and division. I always wondered why modulus is not a basic arithmetic operation. Is there ...
0
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1answer
31 views

Tricks to simplify basic arithmetic expressions?

I am doing a problem set and have several formulas that are quite ugly such as $$b=\left(\frac{3p_b}{2p_r}\right)^{\frac{1}{\rho-1}} \left(\frac{m}{p_r + p_b ...
3
votes
3answers
76 views

A doubt concerning the fundamental theorem of arithmetic

Will a prime $p^{0}$ be considered a unique prime in prime decomposition of a number? If the answer to the above question is yes then it breaks the uniqueness which the Fundamental Theorem of ...
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3answers
2k views

Daily exercises to speed up my mental calculations?

When I was a kid in school my father prevented me from using a calculator when solving my math homeworks. However at that time I was not convinced as of why not to use such a useful tool! So I kept on ...
61
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5answers
15k views

Why is 987654321/123456789 = 8.0000000729?

Many years ago, I noticed that $987654321/123456789 = 8.0000000729\ldots$. I sent it in to Martin Gardner at Scientific American and he published it in his column!!! My life has gone downhill since ...
4
votes
1answer
74 views

Time and distance: Police and a thief with a twist.

A thief was given a head-start of 15 hour. The velocity of the thief being 4 km/hr and the police chasing after him be 5 Km/hr. A dog is moving to and fro between the police and the thief, starting ...
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4answers
153 views

Is there anything (even something weird or fancy) that you can multiply by zero and not get zero?

I'm wondering if there's any kind of "imaginary anti-grassmann" (for lack of a better idea) or some strange object or other in math that you can multiply by zero and somehow not get something other ...
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1answer
32 views

Tree addtion has to do with Pascal's Triangle, why?

Let me define tree addition of a list of numbers as follows: 4 3 2 1 7 5 3 12 8 20 I conjecture that it is true that the tree addition of n numbers ...
3
votes
3answers
68 views

Find the largest $k$ such that $3^k$ divides the product of the first $100$ odd integers

Let $P$ be the product of the first 100 positive odd integers. Find the largest integer $k$ such that $P$ is divisible by $3^k$. There are $50$ odd numbers and $50$ even numbers between $0$ and ...
6
votes
5answers
141 views

Show that $2^{105} + 3^{105}$ is divisible by $7$

I know that $$\frac{(ak \pm 1)^n}{a}$$ gives remainder $a - 1$ is n is odd or $1$ is n is even. So, I wrote $ 2^{105} + 3^{105}$ as $8^{35} + 27^{35}$ and then as $(7\cdot 1+1)^{35} + (7\cdot ...
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3answers
28 views

Complex values of the cube root

I just learned that the cube root has 2 complex roots. For example, the cube root of 8 has : 2 , -1 plus or minus square root of 3 *i I was wondering, how do you find those conjugate complex values ...
0
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2answers
40 views

simplifiying an expression $(n + 1)! − 1 + (n + 1) \cdot (n + 1)!$

I've been stuck on this one problem and I have a problem on the process simplifying this equation so that it is $(n + 2)! − 1.$ $$(n + 1)! − 1 + (n + 1) \cdot (n + 1)!$$ If anyone could shed some ...
35
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6answers
3k views

Why $\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2$?

I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\ \\ \frac1{\sqrt{-1}} &= \frac1i \\ \\ \frac{\sqrt1}{\sqrt{-1}} ...
0
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1answer
24 views

10% fraud, while purchasing and selling. Whats the overall profit?

Ok, so the answer i find logical is $21$%. Like : 100 bucks paid, 110 items got. (10% profit). Then, 110 items you sell at 10% profit, you get 121 items worth of bucks. So, 100 bucks investment, 121 ...
2
votes
3answers
52 views

Number in tens place

A number in tens place in result of $4^{2015} \cdot 9^{2016}$ is? Obviously without using calculator, though I doubt it could count with those high numbers. By tens place I mean, for example if you ...
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vote
2answers
46 views

Number of boys in school

We have $400$ students in a school. Every $20^{th}$ student failed at the end of the school year. Which was $2\%$ of schools girls and $10\%$ of schools boys. The number of all boys attending the ...
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2answers
68 views

Number addition riddle

I got this math "riddle" in one of my math test, and I would love to know how to solve it. If $$S = 1 + 2 + 3 + 4 + \ldots + 2015,$$ then a sum of $$1 + 2 + 3 + \ldots + 2015 + 2016 + \ldots + 4030$$ ...
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1answer
35 views

Number of apples in a basket riddle

You have six baskets with apples - 10,12,15,20,22,25 (this is how many apples there were in them - 10 in first, 12 in second..). Some of the apples are red and some are green. After one basket was ...
0
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1answer
37 views

What is the difference between the largest and smallest possible positive roots?

I am faced with the following question: What is the difference between the largest and the smallest possible positive roots of $4x^5 + 3x^3 -5x^2 + 7x - 12$? Now, my first attempt was to try ...
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3answers
18 views

Is it possible to convert fraction to decimal using only addition and subtraction?

I am working on a programming challenge that requires me to implement addition, division, and modulo using only addition and subtraction. Cool, simple enough: ...
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3answers
82 views

Easy inequality going wrong

Question to solve: $$\frac{3}{x+1} + \frac{7}{x+2} \leq \frac{6}{x-1}$$ My method: $$\implies \frac{10x + 13}{(x+1)(x+2)} - \frac{6}{x-1} \leq 0$$ $$\implies \frac{4x^2 -15x-25}{(x-1)(x+1)(x+2)} ...
2
votes
4answers
628 views

Problem related to a clock

I faced the following problem: At what time after 4 o'clock, the hour and the minute hand will lie opposite to each other? $\quad$ 4-50'-31" $\quad$ 4-52'-51" $\quad$ 4-53'-23" ...
0
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5answers
77 views

Arithmetic Progression.

Q. The ratio between the sum of $n$ terms of two A.P's is $3n+8:7n+15$. Find the ratio between their $12$th term. My method: Given: $\frac{S_n}{s_n}=\frac{3n+8}{7n+15}$ ...
0
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2answers
45 views

Generalized formula for sum of products.

Q:The sum of all possible products of the first n natural numbers taken two by two is? I did not understand the question as it is.What exactly is being asked?I'd really appreciate an answer ...
0
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1answer
21 views

Find the kind of progression

The four positive numbers a,b,c,d are in arithmetic progression.What is the progression sequence of abc,abd,bcd? I found out the common difference b-a,c-b.. but that does not seem to be of much use.
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1answer
42 views

Number of the term

If the sum of n terms in AP is $3(n^2)+5$.What is the number of the term which equals $159$? My attempt: $3(n)^2-3(n-1)^2=159$.I got $n=27$ but the answer given is $21$.
2
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0answers
122 views

Minimum number of real multiplications to multiply two quaternions [closed]

Karatsuba multiplication of two complex numbers can be performed with just three real multiplications (instead of four) as follows: $$(a+bi)(c+di) = (ac-bd) + i ((a+b)(c+d) - ac-bd)$$ We only need the ...
3
votes
2answers
25 views

Calculating the value of numbers with different operations

Calculate the value of: $$-14 + 49 \times 21 - 63 + 56 \div 35 \div 28 \times 70 - 42 \div 7$$ I noticed the numbers are a factor of $7$, so I took out $7$ as a common factor: $$7[-2 + (7 \times 3) ...
0
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1answer
19 views

How to Normalize the Sum of Two Gaussians

I have the following function: $I(\theta_i) = I_0 + I_1\exp(\mu(\cos(\theta_i - \theta_s) - 1))$. Suppose I have two implementations of this function, whose parameters match with the exception of ...
2
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3answers
28 views

Theory behind multiplying decimals

When multiplying two decimal numbers, you first ignore the decimals, find the product, then count the number of decimal places that need to be in the answer by taking the sum of the original decimal ...
2
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1answer
34 views

How do I find the sum of first N numbers common to 2 APs?

Here is the question - Certain numbers appear in both arithmetic progressions 17, 21, 25, ... and 16, 21, 26, ... . Find the sum of first 100 numbers appearing in both progressions. The ...
5
votes
1answer
69 views

Find prime numbers $p,q$ such that $p^n+p^{n-1}+…+p+1=q^2+q+1$

Let $p,q$ are prime numbers and $n$ is a even number such that : $p^n+p^{n-1}+...+p+1=q^2+q+1$ Find $p,q$? I think : $p^n+p^{n-1}+...+p+1=q^2+q+1\Rightarrow p^n+p^{n-1}+...+p=q(q+1)\Rightarrow ...
2
votes
1answer
55 views

Small integral representation as $x^2-2y^2$ in Pell's equation

Let $k$ be a "representable" positive integer, in the sense that $k=|x^2-2y^2|$ for some integers $x,y$. Does it necessarily follow that $k$ can also be represented with small parameters, i.e. ...
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1answer
23 views

Does using the syntax X%% make sense?

I know percentages can be multiplied, as they're basically just fractions, so it makes sense to ask what 50% of 72% of 10 is, for example. But would anybody use an expression like 3%% as shorthand for ...
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0answers
26 views

Simple Interest Problem Ambiguity in Conventions

I am solving some simple interest problems. Following questions are creating ambiguity with conventions, hope someone will clarify what is going on. In what time does sum of money become 4 times ...
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1answer
41 views

How to count the number of integers with a fixed leading digit which are less than a given number?

Consider the numbers $1$ to $5000$. Numbers starting from the digit $2$ would be $2$, $20-29$, $200-299$, $2000-2999$; total would be $1111$. How can one derive a formula for the same? The first ...
3
votes
1answer
24 views

Significant figures during intermediate steps in a calculation

If I have some values to use in a calculation, which all have 3 significant digits, then I know that the result will also have no more than 3 significant digits. Am I allowed to round up/down to 3 ...
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vote
1answer
86 views

What is the difference (or relationship) between geometric length and arithmetic numbers?

In Abbott's Understanding Analysis there was a phrase like, "Ancient Greeks did not understand the difference (or relationship) between geometric length and arithmetic numbers." What is this ...
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2answers
56 views

Multiplication problem.

What would be the value of z in this question? If $z=2,$ the relation becomes $22\cdot wx = 594,$ which gives $wx=27.$ Partial product of $22\cdot 27$ is $154 + 440.$ It's incongruous with the ...
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0answers
32 views

Arithmetic progressions in subset

Let $S$ be a subset of $\{1,\dots,n\}$. Does there exist a good algorithm to find a partition of $S$ into "reasonably long" arithmetic progressions? Many thanks!
3
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2answers
57 views

Find prime numbers $p,q$ such that: $pq| p^p+q^q+1$

Le $p,q$ be prime numbers such that: $pq| p^p+q^q+1$ Find $p,q$ I don't have any ideas about this problem :( Thanks :)
2
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4answers
41 views

$10$ Distinct Integers from a set and their sum equals to $954$

$10$ distinct integers from the set $ \left \{1;2;...;100 \right \} $ are chosen such that their sum is $954$. What is the smallest of the $10$ integers? How do I start this question? I have no idea ...
0
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0answers
28 views

Calculate new least common denominator

A parcel (lot) has an area of $547$ with co-owners that own a certain share in factions: Person $1$ has $\frac{1}{8}$, Person $2$ has $\frac{1}{8}$, Person $3$ has $\frac{2}{8}$, Person $4$ has ...
2
votes
2answers
190 views

Floating point binary arithmetic question

I'm doing a basic class on computer architecture and we dwell into Floating Point Arithmetic, I'm not looking for someone to solve my homework, I'm actually just going through old exams and I'm kinda ...
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2answers
282 views

Long division: 24158 divided 6

Long division has always been a weakness of mine and some how I've gotten through school and sixth form without it, but i'd like to learn it, it's just that I have a problem with intuition. So I know ...
2
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0answers
35 views

How to reduce exponentiation expressions?

It is a simple question but I am afraid of its simplicity. Is that correct : $2^{30}+2^{30}+2^{30}+2^{30} = 2^{30}(1 + 1 + 1 + 1) = (2^{30})\cdot 4 = 2^{30}\cdot2^2 = 2^{32}$? I am doing complex ...