#euclids division algorithm

[jackhkeynes]

Cambrick Codices

summaries of two twentieth-century works of writing which were originally written in one or another standard of the Welsh language.

Children of the Battle of Rexam [Wrexham, 1] is a sensation trevold [hit novel] written by Jon Paldreth and published in 1931. It is a fictionalised account of the orphaned children of the Duke of Chester in the decades following his death at the 1483 Battle of Rexam. By Paldreth's account, the children met practically every famous personage in contemporary western Europe, from General daus Sanz to King Andrew of Markland (in fairness, the eldest child was indeed briefly a hostage of the then-prince Andrew). Moreover, Paldreth contrives to have the children present at almost every significant historical event of the time; unlikely as this all is, the book has enjoyed much use in school history lessons as an approachable introduction to the politics of the time.

The Grimmer Hall [2] is a scholastic reference quire about developments in early methodics [programming]. It was published in 1999 and written by New Leudong methodist Cai Sonquo during his tenure at the University of Mehannin [Brisbane]. The work covers the use of algorithms from antiquity (for example Euclid's method for finding coarsest division), through the early modern period (e.g. the waveform display [Fourier transform] and its application to astronomy), and into the titular "grimmer halls" (rooms of many people—usually women—performing calculations longhand) of the early twentieth century. The work ends its account just before the invention of the entirely mechanical general-purpose grimmer [computer].

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[1] released in Merch as Chulder oth Feight at Rexam /ˈtʃʊl.də‿ɾʊð fɛʃt ət ˈɾɛk.səm/, and originally published in Welsh as Pa Ðywdan y Plant Car Vantel /pa ðəuˈdɔn ə plant kaː vənˈtɛɬ/ "What the Children of Rexam Said"

[2] originally in Welsh as Łis y Grimrageð /ɬiːs ə ˌgɾɪm.ɾəˈgɛð/ "The Computing-Women's Hall".

[vasanthasworld]

CBSE Solutions For Class 10 Mathematics Chapter 1 Real Numbers

[jonathankatwhatever]

It’s 20 Feb 2025, and I’m working through some search issues which I doubt I’ll be able to put into words. I found myself thinking about SAT3 in the middle of the night. I view that as an Extent, as a series of Triangulars, and thus you see Hexagonal, meaning each burst of 3, each this or this or this connects to the other, meaning 3 or’s connected by and. This is why SAT3 is NP-complete and why NP problems reduce to it.

Various thoughts occur, like ‘we have the computing power to check Extents, and thus to factor the space, any space. We have an infinite focus capability, so our issues are where we should look and what we should be looking for, which is in many ways the same as saying we need to identify the level we need to see. That’s why we’re at this level, and you should be able to see that we’re connecting the layers in D-structure so the structure speaking this way is exactly how this structure can speak, through the revelation, both in the mystic and practical sense, yes a gsRealization is one which combines those because those are what embeds in your fabric as myths. Yes, so that explains the need for Lincoln and why changing roles are perceived inappropriately as absolutes: so you can justify destruction of the past to eliminate pain you feel inside your head, meaning you continue to impose external solutions when the actual solution is internal.’

If you’re answering questions, why this algorithm thinking? This is clearly big O exponential because you can literally see the Extent has 2 branches and thus choice is 2^x. That is why we call this Halving/Doubling. And why 2 is prime.

I realized yesterday that gsPrimes unveils cool stuff. Like Sylow subgroups are an implementation of gsPrimes. Makes it much easier to grasp what’s happening when you see the structures.

Oh, okay, I forgot to say that in gs, the same counting along the Extent is of course along the szK. That’s part of the mapping of the reals. Can you define the irrationals a bit better? I get the coming together along the szK because that’s literally the root of any n, something we drew out years ago. How does that distinguish from rationals? All rationals are at corners, and since the corners are Bips at the other layers, then the corners are Ends with the axis flavor, meaning they’re xyK. So take some random but small enough to see division, like 33/2, and that maps how? Think 33 long, and 2 of those meaning 66, so the ruler max is 33 and there’s 1 of those. So 33 has a ruler of 16, thinking like Euclid, and the 1 left over Halves as known. That’s all by manipulation of gs in layers. Now generate the szK as the hypotenuse: you need to define a 2Square to define the root of that, and that involves a twist.

Oh, I’m starting to remember a bit more. I forget this all occurred and that it acts like an answer book which connection to the Actuality of you reveals. So that comes up. How does that work? The logic is fairly clear, right? You connect to a shared Thing and thus to an End which pairs in Triangular, and that Triangular forms an Extent, which gives a lot of meaning to all those signs, and Actuality is the D3-4Space realization of that, meaning the 0-1-0 mechanism, which I tend to call renormalization because that fits the concept of ‘here’s this gsProcess which covers this gap between D3-4 Things, and that is true over any physical distance because this means over the Boundary.

I have to run errands. I’m concerned that she is not accepting there’s a physical issue, though she falls often. Actualities are complicated.

Not quite ready to stop.

Okay, we were talking about number forms again. Maybe we can remember them this time. Transcendental is what? A lot of gsProcess, and involves CR, right? Irrationality means a shift to 1Square. I think transcendental means a CR shift of 1Square. Take e: why is it transcendental? It’s associative over the count of 1 to 2, and associative means gsPairings, which means the entire structure CR’s to count within gs, meaning to map to Actuality, often as a process. This fits for Catalan’s constant. If we look at Liouville’s numbers, we see the process. But first, we left out algebraics, which define the amount and type of gsProcess allowable for the irrational category.

I’ll finish the phrasing later. The idea is there.

[4denthusiast]

The trial and error in the first one is basically avoidable. Working in the integers mod 7,

60n+1 = 0

4n+1 = 0

4n = -1

n = -1/4

Then since the multiplicative inverse of 4 mod 7 is 2 (which is probably most easily found just by checking all the cases, but that's much quicker than doing the calculation with 60), n=-2 (mod 7) and the smallest possible positive value of n is 5, giving 60*5+1=301. This is basically the first step of Euclid's algorithm for modular division, but if the numbers were big enough that the full algorithm were necessary, that might take a bit too much memory to do in your head.

I bought this book from goodwill for a dollar and a lot of the games are hit or miss but overall I'm enjoying it, but there is a section of mental games (just to be played in your head, no pen or paper), and this one game is so unbelievably hard??

Note that you are supposed to do these entirely in your head, writing nothing down.

I wasn't timing myself because I was working on these while I was falling asleep last night but I think the second one took me over an hour.

1. A certain number leaves a remainder of one when it is divided by 2, 3, 4, 5 or 6 but leaves no remainder when it is divided by 7. What is the smallest possible value of the number?

2. A certain number, when divided by 2, 3, 4, 5, 6, 7, 8, 9 and 10, leaves remainders of 1, 2, 3, 4, 5, 6, 7, 8 and 9 respectively. What is the smallest possible value of the number?

And there are 20 of these questions. Totally worth the dollar, hours of fun (if you are the type of nerd I am)

[edusquaremaths]

NCERT Class 10

Chapter: Real Numbers

Exercise 1.1

Question 3: An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

[lawsofemotion]

Updated Number Theory! I finished the notes for Chapter 1. However, there are no practice problems or 1.4. The practice problems will come later next year. Have a happy Christmas and holidays!

[qhd4]

I never liked Euclid's algorithm all that much, the results that come from it are much more interesting then iterated division, plus the proofs that use it are ugly imo, a bunch of tedious calculation.

Jacobson's Basic algebra 1 gives such a nice argument for why algebraic numbers have minimum polymials. Consider a field extension E/F (i.e. a field E that contains a field F). For any u ∈ E we have in E the subring F[u] and the subfield F(u). These are defined as the smallest subring and subfield of E respectively that contain both F and u. If E = F(u) for some u ∈ E, then E/F is called a simple field extension.

Let F(u)/F be an arbitrary simple field extension. Consider the polynomial ring F[X]. By the universal property of polynomial rings (which is essentially what one means when they say that X is an indeterminate), there is a unique ring homomorphism F[X] -> F(u) that sends any a ∈ F to itself and that sends X to u. If the kernel of this homomorphism is the zero ideal, then F[u] is isomorphic to F[X], so u is transcendental over F. If the kernel is non-zero, then by definition there are non-zero polynomials p ∈ F[X] such that p(u) = 0 in F(u), so u is algebraic over F. Because F[X] is a principal ideal domain, there is a polynomial p that generates the kernel. In other words, p divides all polynomials q such that q(u) = 0 in F(u). Two polynomials (over a field) generate the same ideal if and only if they differ by a constant factor, so there is a unique monic minimum polynomial in F[X] for u.

[ashishkumarletslearn]

#RealNumbers #Class10Maths

In this lecture I have discussed the Meaning & Method to find HCF / GCD and Euclid's Division Algorithm of Chapter 1 Real Numbers Class 10 Maths.  

[allrounderknowledgehub]

Chapter : Real Numbers

Exercise : 1.1

Question 1. Use Euclid's division algorithm to find HCF of

(i) 135 and 225

(ii) 196 and 38220

(iii) 867 and 255

Solution 1.

(i) 135 and 225

Since 225>135, we can apply Euclid's division lemma to 225 and 135

225 = 135×1 + 90

Since the remainder 90≠0 , we can apply Euclid's division lemma to 135 and 90

135 = 90×1 + 45

Again we can see that the remainder 45≠0 we will again apply Euclid's division lemma to 90 and 45

90 = 45×2 + 0

We get remainder = 0

So in last step divisor is 45 so the HCF of 135 and 225 is 45.

To check full answer click here

https://allrounderknowledgehub.blogspot.com/2021/05/ncert-solutions-class-10th.html?m=1

[wisdomrays]

TAFAKKUR: Part 241

MUSLIM CONTRIBUTIONS TO MATHEMATICS: Part 1

When we talk about Muslim contributions to mathematics we are usually referring to the years between 622 and 1600 ce. This was the golden era of Islam when it was influential both as a culture and religion, and was widespread from Anatolia to North Africa, from Spain to India.

Mathematics, or "the queen of the sciences" as Carl Friedrich Gauss called it, plays an important role in our lives. A world without mathematics is unimaginable. Throughout history, many scholars have made important contributions to this science, among them a great number of Muslims. It is beyond the scope of a short article like this one to mention all the contributions of Muslim scholars to mathematics; therefore, I will concentrate on only four aspects: translations of earlier works, and contributions to algebra, geometry, and trigonometry. In order to understand fully how great were the works of scholars in the past, one needs to look at them with the eye of a person of the same era, since things that are well-known facts today might not have been known at all in the past.

There has never been a conflict between science and Islam. Muslims understand everything in the universe as a letter from God Almighty inviting us to study it to have knowledge of Him. In fact, the first verse of the Qur'an to be revealed was:

Read! In the Name of your Lord, Who created… (Alaq 96:1).

Besides commanding us to read the Qur'an, by mentioning the creation the verse also draws our attention to the universe. There are many verses which ask Muslims to think, to know, to learn and so on. Moreover, there are various sayings of the Prophet Muhammad, peace be upon him, encouraging Muslims to seek knowledge. One hadith says, "A believer never stops seeking knowledge until they enter Paradise" (al-Tirmidhi).

In another hadith, the Prophet said, "Seeking knowledge is a duty on every Muslim" (Bukhari). Hence it is no surprise to see early Muslim scholars who were dealing with different sciences.

TRANSLATIONS

Prophet Muhammed (pbuh) said, “Knowledge is the lost property of a Muslim; whoever finds it must take it” ; hence Muslims started seeking knowledge. One way they did this was to start translating all kinds of knowledge that they thought to be useful. There were two main sources from which Muslim scholars made translations in order to develop the field of science, the Hindus and the Greeks. The Abbasid caliph al-Mamun (804–832) had a university built and ordered its scholars to translate into Arabic many works of Greek scholarship. Between 771 and 773 CE the Hindu numerals were introduced into the Muslim world as a result of the translation of Sithanta from Sanskrit into Arabic by Abu Abdullah Muhammad Ibrahim al-Fazari. Another great mathematician, Thabit ibn Qurra, not only translated works written by Euclid, Archimedes, Apollonius, Ptolemy and Eutocius, but he also founded a school of translation and supervised many other translations of books from Greek into Arabic. While Hajjaj bin Yusuf translated Euclid’s Elements into Arabic, al-Jayyani wrote an important commentary on it which appears in the Fihrist (Index), a work compiled by the bookseller Ibn an-Nadim in 988. A simplified version of Ptolemy’s Almagest appears in Abul-Wafa’s book of Tahir al-Majisty and Kitab al-Kamil. Abu’l Wafa Al-Buzjani commented on and simplified the works of Euclid, Ptolemy and Diophantus. The sons of Musa bin Shakir also organized translations of Greek works.

These translations played an important role in the development of mathematics in the Muslim world. Moreover, the ancient Greek texts have survived thanks to these translations.

ALGEBRA AND GEOMETRY

The word "algebra" comes from "Al-Jabr", which is taken from the title of the book Hisab Al-Jabr wal Muqabala by Muhammad ibn Musa al-Khwarizmi (780–850). Al-Khwarizmi, after whom the "algorithm" is named, was one of the great mathematicians of all times. Europe was first introduced to algebra as a result of the translation of Khwarizmi's book into Latin by Robert Chester in 1143. The book has three parts. The first part deals with six different types of equations:

(ax2 = bx) ; (ax2 = b) ; (ax = b) ; (ax2 + bx = c) ; (ax2 + c = bx) ; (bx + c = ax2)

Khwarizmi gives both arithmetic and geometric methods to solve these six types of problems. He also introduces algebraic multiplication and division. The second part of Hisab Al-Jabr deals with mensuration. Here he describes the rules of computing areas and volumes. Since Prophet Muhammad, peace be upon him, said, “Learn the laws of inheritance and teach them to people, for that is half of knowledge," the last and the largest part of this section concerns legacies, which requires a good understanding of the Islamic laws of inheritance. Khwarizmi develops Hindu numerals and introduces the concept of zero, or “sifr” in Arabic, to Europe. The word “zero” actually comes from Latin “zephirum,” which is derived from the Arabic word “sifr.”

The three sons of Musa bin Shakir (about 800–860) were perhaps the first Muslim mathematicians to study Greek works. They wrote a great book on geometry, Kitab Marifat Masakhat Al-Ashkal (The Book of the Measurement of Plane and Spherical Figures), which was later translated into Latin by Gerard of Cremona. In the book, although they used similar methods to those of Archimedes, they move a step further than the Greeks to consider volumes and areas as numbers, and hence they developed a new approach to mathematics. For example, they described the constant number pi as “the magnitude which, when multiplied by the diameter of a circle, yields the circumference.”

A well-known poet, philosopher and astronomer Omar Khayyam (1048–1122) was at the same time a great mathematician. His most famous book on algebra is Treatise on the Demonstration of Problems of Algebra. In his book besides giving both arithmetic and geometric solutions to second degree equations he also describes geometric solutions to third degree equations by the method of intersecting conic sections. He also discovered binomial expansion [26]. His work later helped develop both algebra and geometry.

Thabit bin Qurra (836–901) was an important mathematician who made many discoveries in his time. As mentioned in the Dictionary of Scientific Biography he “played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-Euclidean geometry. In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics.”

To give an idea of his importance, we will just give here, without details, one of his theorems on amicable numbers. Two natural numbers m and n are called “amicable” if each is equal to the sum of the proper divisors of the other:

for n > 1, let pn=3.22n–1 and qn=9.22n–1–1. If pn–1 , pn and qn are prime numbers, then a=2n pn–1 pn and b=2nqn are amicable.

[ncertsolutionsbook]

What is Euclid Division Algorithm

What is Euclid Division Algorithm Euclid’s Division Lemma: For any two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, where 0 ≤ r < b. For Example (i) Consider number 23 and 5, then: 23 = 5 × 4 + 3 Comparing with a = bq + …

[learnenglishcbsein]

Class 10 Maths NCERT Solutions Chapter 1 Real Numbers

Class 10 Maths NCERT Solutions Chapter 1 Real Numbers

Class 10 Maths NCERT Solutions Chapter 1 Real Numbers Class 10 Maths Real Numbers Exercise 1.1

Question 1 Use Euclid’s division algorithm to find the HCF of : (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255 Solution: (i) 135 and 225

Method 1:

Method 2:

(ii) 196 and 38220 Method 1:

Method 2:

(iii) 867 and 255 Method 1:

Method 2:

Question 2: Show that any positive odd integer is of the form 6q…

[learncbse-science]

Class 10 Maths NCERT Solutions Chapter 1 Real Numbers

Class 10 Maths NCERT Solutions Chapter 1 Real Numbers

Class 10 Maths NCERT Solutions Chapter 1 Real Numbers Class 10 Maths Real Numbers Exercise 1.1

Question 1 Use Euclid’s division algorithm to find the HCF of : (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255 Solution: (i) 135 and 225

Method 1:

Method 2:

(ii) 196 and 38220 Method 1:

Method 2:

(iii) 867 and 255 Method 1:

Method 2:

Question 2: Show that any positive odd integer is of the form 6q…