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

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How to define multiplication in $\mathbb{Z}$ with divisibility and addition?

Q: Show that $(\mathbb{Z},|,+,0,1)$ defines multiplication in $\mathbb{Z}$. I know how to do this in $\mathbb{N}$, but I'm stuck trying to do this is $\mathbb{Z}$. The idea I have is to define lowest ...
2
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3answers
61 views

Comparing the size of $(\sqrt{5})^e$ and $e^{\sqrt{5}}$

So I have to figure out which one is bigger between $(\sqrt{5})^e$ and $e^{\sqrt{5}}$. After some trial and error I've come to the conclusion that $(\sqrt{5})^e > e^{\sqrt{5}}$. But of course I ...
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0answers
13 views

Finding the frequency of a fan using a slow motion camera

This problem has me a little stumped. I'm not sure if my answer is correct and would just like to check: The camera shoots at 187 frames per second. The fan takes 33 frames to complete one revolution ...
0
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1answer
30 views

$b^{\frac{m}{n}}=(b^{\frac{1}{n}})^m=(b^m)^{\frac{1}{n}}$ except $b$ is not negative when $n$ is Even.

The following property, known as Rational number property, is taken from the book (I am following now a days) College Algebra by Raymond A Barnett and Micheal R Ziegler I restate, ...
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2answers
66 views

How to find minutes?

Need help solving this real life problem, I have an SD Card of $4$GB(gigabyte), and a $32$ second video occupies $6.12$MB(megabyte), I need to know how many minutes or seconds can this $4$GB SD Card ...
2
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2answers
63 views

Variations in the successor fuction from Peano's axioms

Concerning the successor function in Peano's axioms, what prevents me from defining it in the following way: 0 to 2, 2 to 1, 1 to 4, 4 to 3, 3 to 6, 6 to 5, 5 to 8, 8 to 7 ... and so on. It seems ...
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2answers
55 views

Simplifying Square Roots Frustration

Okay, I'm really frustrated with this. So, when you have $3 \sqrt 5 + 5 \sqrt 5$, you get $8\sqrt5$, right? Okay, so what do I do for here: $\sqrt{11} - 3 \sqrt{11}$ Is it just $-3 \sqrt{11}$ ? ...
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1answer
28 views

Interval arithmetic - faster version

As per the below question picked from self training exercise: Q4: In passing, Ben also cryptically comments, "By testing the signs of the endpoints of the intervals, it is possible to break ...
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1answer
43 views

order of operations in different cultures?

Are there any cultures or countries around the world that use a different convention for order of operations than the BEDMAS convention? i.e.: Parentheses Exponents & Roots Multiplication & ...
2
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2answers
59 views

How many numbers smaller that $N$ can be written as a sum of two square numbers?

We define $$a_N =\# \{ n \leq N, \exists (n_1,n_2) \in \mathbb{N}^2, n = n_1^2 + n_2^2 \}.$$ Can we have the exact value of $a_N$, or at least an asymptotic behavior such as $$ \alpha N \leq a_N \leq ...
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0answers
18 views

solving Multi operation equations correctly

Is it possible to solve an equation with different operations in it correctly without using orders of operation? I was having a discussion with my friend who believes you can solve an equation from ...
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8answers
155 views

Which is larger, $\sqrt{3} + \sqrt{5}$ or $\sqrt{2} + \sqrt{6}$?

The clue given by the text is to "use the fact that $\sqrt{x}$ is increasing." I was able to get the correct answer here by squaring both expressions. But I don't think I made use of the text-prided ...
0
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1answer
35 views

Profit-Loss : Discount

On an order of 5 dozen boxes of a consumer product, a retailer receives an extra dozen free. This is equivalent to allowing him a discount of: 15% 97/6% 50/3% 20% I don't know how to put these ...
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1answer
26 views

Profit and Loss : Manufacturer

A manufacturer undertakes to supply 2000 pieces of a particular component at Rs.25 per piece. According to its estimates, even if 5% fail to pass the quality tests, then he will make a profit of ...
3
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1answer
49 views

Finding all possible pairs of positive integer values

The ratio of the sum of two positive integers to their difference is $7:5$. If the the sum of the two numbers is at most $25$, find all possible values for the pair of numbers. Let $m$ be the first ...
2
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3answers
78 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
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1answer
23 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 ...
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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
51 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 ...
0
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1answer
43 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 ...
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3answers
80 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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1answer
77 views

Which one of the following logical propositions is to be preferred?

I'm trying to update the symbolism of Giuseppe Peano's "Arithmetices Principia", to make the translation freely available. Might I ask you, which of the following might be a correct mathematical ...
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1answer
184 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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1answer
33 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 ...
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3answers
72 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 ...
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3answers
35 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 ...
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2answers
41 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 ...
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1answer
41 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 ...
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5answers
146 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
59 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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2answers
50 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
82 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
66 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 ...
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1answer
147 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
20 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)} ...
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2answers
75 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
27 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
43 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 $21$.
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10answers
405 views

Prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$

I'm trying to prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$ I've first written down $-5\lt a-b \lt5$ and have tried to add different things from all sides of the ...
2
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3answers
42 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 ...
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1answer
42 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 ...
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1answer
80 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 ...
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1answer
71 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 ...
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1answer
60 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
25 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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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) ...
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46 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
34 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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1answer
116 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 ...