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

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hard question, please help [on hold]

11) Assume a sorted array (A) of size n. Propose an algorithm for finding two elements x and y in A that minimize |x-y|. Your algorithm should run in O(n) time for full credit. (Note: |x-y| represents ...
4
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0answers
37 views

Extention of Euclid's GCD Algorithm. (The Art of Computer Programming, Volume 1, Edition 3, Section 1.2.1, Exercise 12)

Euclid's GCD algorithm which is used to find GCD of two input numbers, say, $c$ and $d$, needs the inputs to be positive integers. Exercise 12 provides an extension to this algorithm and allows $c$ ...
-6
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0answers
86 views

How to solve this horizontal multiplication $234\cdot 345\cdot 542 =?$ [on hold]

Kindly solve the following problem step-by-step How to solve this horizontal multiplication $234\cdot 345\cdot 542 =?$ Yours sincerely, RAJI REDDY. K
-6
votes
0answers
29 views

formulate a plan, add, subtract, divide [on hold]

An elevator began on the 4th floor of a building. From there it the elevator traveled 8 floors up, then 3 floors down, then 1 floor up. On what floor did the elevator stop?
1
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2answers
24 views

Check if $(a_n)$ is arithmetic progression with a given $S_n$

We have $(a_n)$ with $S_n = n^2$. Check if $(a_n)$ is an arithmetic progression or not. NOTATION $$(a_n)_{n\ge1}\\S_n = a_1 + a_2 + \cdots + a_n, \forall n\in \mathbb{N}^*$$
1
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1answer
18 views

Are Primitive Dirichlet Characters linearly independent.

For a positive integer $N$, let $$S_N=\{ \chi~\mid~ \chi \text{ is primitive Dirichlet characters modulo }F,\text{ where } F\mid N \}.$$ I want to check the Linear independence on $S_N$. More ...
0
votes
1answer
20 views

How to get the maximum and minimum number of length $m$ and the sum of the digits $s$

How to get the maximum and minimum of length $m$ and the sum of the digits $s$ By example: Length: 2 Sum of its digits: 15 Max: 96, Min: 69 Length: 2 Sum of its digits: 2 Max: 20, Min: 11
5
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1answer
96 views

Set with distinct subset sums

The problem is as follows : Given a set A with distinct positive integer elements, prove that there always exists another set B consisting of positive integers, s.t., The size of B is less than or ...
6
votes
0answers
30 views

Product of permutations of consecutive numbers yields arithmetic sequence

Let $n\geq 3$ be an integer, and $a,b$ be positive integers. Let $c_1,\ldots,c_n$ be a permutation of $a,a+1,\ldots,a+(n-1)$, and $d_1,\ldots,d_n$ be a permutation of $b,b+1,\ldots,b+(n-1)$. Is it ...
3
votes
1answer
37 views

How to calculate (predict) how much I will be paid in 30 days if I know how much I was paid in the first 9?

I'm a very simple man living his life. I don't know much math. This is a real world scenario of me trying to apply math and trying to find how much approximately I will be paid this month. I know some ...
0
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1answer
24 views

Why multiplying or dividing a positive value by a negative value gives a negative answer? [closed]

As in the question why should it be negative? What is the reason?
0
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1answer
38 views

Arithmetic series $1 + 3 + 5 + \dots + (2n + 1)$

$$1 + 3 + 5 + \dots + (2n + 1) $$ For the above question, the answer is $(n + 1)^2$ and I understand that $n$ is the number of terms. If I let my $n$ is $3$, that means I add $1 + 3 + 5 = 9$ but if I ...
-1
votes
1answer
54 views

What happens when we divide zero by zero? [duplicate]

I know that 0/any number=0 and any number/0=no answer, (negative) infinity, undefined. So, what happens if we divide zero by itself? Is it zero? Is it infinity? Well, infinity isn't a real number, ...
1
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1answer
44 views

$2013!$ ends in a string of zeros. How many of them are there?

$2013!$ ends in a string of zeros. How many of them are there? Work The answer is given like this:(Kindly explain why this is so) $2$’s and $5$’s pair off to produce multiples of 10 and since there ...
2
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1answer
36 views

Dropping the “Lowest Grade” Problem

Let $A=\{a_1, a_2, ... , a_n\}$ be a set of non-negative real numbers and $B=\{b_1, b_2, ..., b_n\}$ be sets of positive real numbers. Let $s = \dfrac{ \sum_A a}{\sum_B b} = ...
1
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2answers
48 views

Understanding the concept of multiplication

Since multiplication is such a basic algorithm in math we rarely stop to think about what it really is about, so please, help me understand? When we multiply two numbers we are basically increasing ...
0
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1answer
16 views

Can this simple divisibility property on binomial coefficient be proved without Gauss' lemma?

Consider the following property : ( * ) if $n\geq 1$, then $a_n=\binom{2n}{n}$ is divisible by $2n-1$. One can show that ( * ) is true as follows : $2n-1$ divides $na_n$ (because of the identity ...
-1
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0answers
49 views

In $\triangle ABC$ , find the value of $\cos A+\cos B$ [closed]

The sides of $\triangle$ABC are in Arithmetic Progression (order being $a$, $b$, $c$) and satisfy $\dfrac{2!}{1!9!}+\dfrac{2!}{3!7!}+\dfrac{1}{5!5!}=\dfrac{8^a}{2b!}$, Then prove that the value of ...
3
votes
2answers
68 views

Integer inequality: $x + y +z> a + b + c$ does not imply $xyz > abc$

Prove by contradiction that for any integers $x,y,z,a,b,c$ greater than $0$ such that $x+y>a+b$, it is not implied that $x\cdot y\cdot z>a\cdot b\cdot c$? Obviously this statement is true. ...
3
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3answers
65 views

Ratio between two numbers is 6:7 and the difference between them is 10. What are the two numbers?

I know the numbers are $60$ and $70$ but I got that by trial and error. Is there some other more logical way to do this problem?
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2answers
34 views

show that if n is an odd prime number it is not necessarily true that n+2 is prime [closed]

Have no idea what I'm doing guys. would appreciate a play by play of how to calculate merci
0
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1answer
32 views

Matrix addition

How do I solve the following? [2x1 -3x2 + x3; 4x1 - 2x3] + [x1 +2x2; 0x1 - 2x2; 4x1 + x2]^T When I do the transpose of the second matrix and try to add them together I get lost. Should I consider x1 ...
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3answers
35 views

Let $a = −215$ and $b = 17$. Find the integers $q$ and $r$ with $0 \leq r < b$ such that $a = qb+r$.

Let $a = −215$ and $b = 17$. Find the integers $q$ and $r$ with $0 \leq r < b$ such that $a = qb+r$. I don't know where to stop $$a = qb + r$$ $$-215 = q \cdot (-17) + r$$ help me continue. ...
2
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1answer
29 views

What is the answer for this aptitude question?

If 13!/2^x is an integer, which of the following represents all possible values of x? a) 0 <= x <= 10 b) 0 < x < 9 c) 0 <= x < 10 d) 1 <= x <= 10 e) 1 < x < 10 The book ...
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2answers
33 views

Find closed form for $2 \times (n + (n + 1) + \cdots + (2n - 1))$

How can we find a closed form for this sum: $$2 \times (n + (n + 1) + \cdots + (2n - 1))$$? Example: $$(7+6+5+4)+(4+5+6+7)=\frac{3}{4} 8^2 - \frac{1}{2} 8$$
4
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1answer
46 views

$4^{101} \mod 101$ without Fermat's little theorem

Can anyone tell me how to find the remainder of $\frac{4^{101}}{101}$ without using Fermat's little theorem ? I tried doing $$4^{101} \equiv 2^{202} \equiv (5\cdot 101 + 7)^{22} \cdot 2^{4} \equiv ...
1
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2answers
34 views

If 8 divides $a^2$ + $3b^2$, prove that both a and b are even.

Question: Prove that if $a$ and $b$ are integers and 8 divides $a^2$ + $3b^2$, then both a and b are even. I'm able to prove that 8 divides $a^2$ + $3b^2$ if both $a$ and $b$ are even using the ...
0
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1answer
34 views

Compute the product of digits of P

Give $P$ a integer number where $$P=2^{3^{4^{5^{\dots1000}}}}$$ Then Compute The product of dígits of $P$ Compute $P\pmod{5}$ for The segond i think its will be something like ...
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0answers
10 views

X numbers that when subtracted will produce the same absolute value

Let's say I have X unique numbers and I choose one number y out of this set. Is it possible to create these X numbers such that the absolute difference between y and any other number in X will always ...
1
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6answers
125 views

Are there numbers such that A + B = 10A+B? [closed]

I was just wondering, apart from zero,are there numbers where $A+B=10A+B$ (the number AB)?
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5answers
84 views

Any good exponent calculators? [closed]

Can somebody give me a calculator that can do something like "5000^5000" ? I'm talking about a calculator that can do 2000 to the power of 2000 and not give me E+. Thanks!
0
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2answers
23 views

How to solve a subtraction equation that results in a negative number?

So we're home-schooling our 6-yr old and she's doing basic addition/subtraction/multiplication/division.. and I wrote a subtraction problem backwards by accident the other day, and I can't figure out ...
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1answer
31 views

Describing a discrete dynamic system

Model There are three types of animals: $Y$, young (0-5 years old) $A$, adult (5-10 years old) $O$, old (10 years old or more) The initial conditions of the system are $Y_0=2500$, $A_0=1200$, ...
1
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1answer
68 views

Prove that $\binom{p}{k}/p $ is integral for $k\in \{1,..,p-1\}$ with $p$ a prime number

I started by induction on $k$ For $k=1$ then : $1\in \mathbb{N}$ For $k=2$ then : $\frac{(p-1)}{2!} \in \mathbb{N}$ , indeed for all $p>{2}$, $p-1$ is even. (We still have $k<p$ it's ...
2
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3answers
18 views

arithmetic modulo field with real numbers vector space

if i choose modulo3 to be the field, and the real numbers to be the vector space. how do i Multiplier vector in scalar? for example i take "4" from the vector space of real numbers and want to ...
2
votes
2answers
199 views

Cube roots answers

So I am in a middle of a problem and I got stuck at cube root of $8$. I know the answer is is $2$ but my book is showing a positive and negative 2. I thought that cube roots had only one answer. ...
1
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2answers
26 views

How is $P=4P−3RP$ equal to $1=4−3R$

How is $P=4P−3RP$ equal to $1=4−3R$ when by dividing both sides by $P$ one eliminates two $P's$ at the right side of the equation while having only one $P$ in the denominator.
11
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8answers
437 views

Make the number $100$ out of $1,2,3,$ and $4$ digits, without repeats

How can we make the number $100$, using only the following digits: $1,2,3,4$. You cannot repeat any of them.
0
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4answers
63 views

if $a+ b = 9 $and $ab = 1$. what will be the $a^3 + b^3 =$?

How can I solve this? Or, is it given properly? If $a + b = 9$ and $ab = 1$. What is $a^3 + b^3 = $?
33
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10answers
5k views

Is it possible to simulate a floor() function with elementary arithmetic?

I'm using a "programming language" that only allows basic operations: addition, subtraction, multiplication, and division. Is it possible to emulate a floor function (i.e. drop the decimals a number) ...
0
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1answer
28 views

Theorems of GL in modal logic

So I've been reading George Boolos' "The Logic of Provability" and he's explaining different systems of modal logic. He's taken as his basic symbols → (implication), □ (necessity), ⊥ (falsehood), a ...
0
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1answer
17 views

Hexidecimal subtraction when more than one borrow is needed

I wanted to know how I would subtract two hexadecimal values from each other when more than one borrow is needed. Given 0x00000200 - 0x00000004 Here is what I ...
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2answers
47 views

Why does cube rooting a negative number always give us an answer?

Well, I multiply a negative number twice by itself and I always get a negative number! Look:$$-1.5\cdot-1.5\cdot-1.5=-3.375$$$$-6\cdot-6\cdot-6=-216$$I think this is why. I also know that the number ...
0
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2answers
34 views

Why does square rooting a negative number never give us an answer?

I used to do this on my calculators and it never worked! I think it's because you can't multiply any number by itself to get a negative number. Is that even true? I think it is! I've tried it out ...
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4answers
104 views

Is $\frac{5x}{3}$ The Same As $\frac{5}{3}x$?

I believe they are the same but I'm not sure. Can someone please clarify this for me, and also explain why it would be the same or different.
2
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1answer
102 views

Do these arithmetic rules work? They extend the number system by a zero not based on the empty set that is a divisor with unique quotients.

These rules are part of an attempt to define an additive identity in terms of division in basic standard arithmetic. The difficulties with defining division by $0$ are well known. In order to ...
2
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3answers
186 views

fast mental arithmetic: is it an algorithm or table-like structure?

edit Removing the fluff, the question is: When solving problem X by heart, how does the mind reaches the solution very fast? by 'running' an algorithm or 'accessing' a table The 'fluffy' version: ...
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2answers
78 views

What is $\frac{9}{3} - \frac{1}{2}$?

I need to compute $\frac{9}{3} - \frac{1}{2}$. I got an answer of $\frac{8}{6}$ but that is incorrect. $\frac{5}{2}$ is the correct answer. How is this possible?
4
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3answers
202 views

Why do I get two different results for the reciprocal of $i$?

I am aware that the correct answer is $$\frac{1}{i}=\frac{1}{i}\frac{i}{i}=\frac{i}{i^2}=\frac{i}{-1}=-i$$ But equally, I find no error here: $$\frac{1}{i}=\frac{1}{\sqrt{-1}}= ...
0
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2answers
31 views

How to show that bound?

How to show that $\prod\limits_{i=1}^k \dfrac{k+i}{4i}$ is less than or equal to $1/2$ for all $k \ge 1$ Integer. I coulden't understand the answer in How to prove the bound on the probability?