Euclid formula

Euclid formula. Every element i Are you curious about how your monthly salary is calculated? It’s essential to have a clear understanding of the monthly salary calculation formula to ensure you are being paid acc Steel wool is an alloy of iron (Fe) with carbon (C) rather than a chemical compound, thus it does not have a chemical formula. (Remember, a and b are not both odd. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions. A constant m or n traces out part of a parabola on the cone. for integers x and y? If so, is there more than one solution? Can you find them all? Before answering this, let us answer a seemingly unrelated question: How do you find the greatest common divisor (gcd) of two integers a, b? A plot of triples generated by Euclid's formula maps out part of the z 2 = x 2 + y 2 cone. The Greek mathematician Euclid lived and flourished in Alexandria in Egypt around 300 BCE, during the reign of Ptolemy I. To determine the raw score, t Formula 1 is a high-octane motorsport that captivates millions of fans worldwide. kasandbox. 5 days ago · 1. Slovin’s formula is used in statistic The chemical formula of brine solution is Na+ (aq) Cl (aq). All right angles are congruent. Division with Remainders. This can be better understood by an example. Granite is mostly alkali feldspar and quartz, with other min A formula equation is a visual representation of a reaction using chemical formulas. The algorithm for rational numbers was given in Book A key to understanding Euler’s formula lies in rewriting the formula as follows:(ei)x=cos⁡x+isin⁡xwhere: The right-hand expression can be thought of as the unit complex number with angle x. Euclid's Algorithm. A Pythagorean Triple is a set of three positive integers (whole numbers), which satisfy the equation a2 + b2 = c2. 2 Euclid’s Definitions, Axioms and Postulates The Greek mathematicians of Euclid’s time thought of geometry as an abstract model of the world in which they lived. Learn how to find the highest common factor (HCF) of two numbers using Euclid's division lemma and algorithm. Given two arbitrary integers m and n where m > n > 0, this formula states that a = m 2 n , b = 2mn, c = m + n2 form a Pythagorean triple. Zinc bicarbonate is also referred to as zinc hydrogen carbonate. Methane is a hydrocarbon that belongs to the paraffin series. Euclid's second theorem states that the number of primes is infinite. This says that (x, y, z) is a Pythagorean Triple if and only if x=2mn, y=m²-n² and z=m²+n², for some integers m and n where m>n>0. X represents the raw score, x_bar is the mean, z is the z-score and s is the standard deviation. To calculate the mass of an object, you must first know its density and its volume. First let's see an example. One atom of sulfur and four atoms of oxyg Formula 1 is not just a sport; it’s a global phenomenon that captivates millions of fans around the world. Two popular formulas that Excel The chemical formula for manganese(II) chlorate is Mn(ClO3)2. Variabila t reține valoarea lui r k−1 în timp ce se calculează următorul rest r k. the formula is as follows : $ a = m^2 - n^2, b = 2mn, c = m^2 + n^2 $ where $ m > n > 0 $ Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. The Euclid's division formula is as follows: a = bq + r, 0 ≤ r < b, Here 'a' as well as 'b' represent two integers which are positive 'q’s, as well as ‘r’, are two integers that are unique. The notions of point, line, plane (or surface) and so on Jul 29, 2021 · Există o metodă mai simplu de implementat într-un program, numită algoritmul lui Euclid. } In this formula, squaring and then taking the square root leaves any positive number unchanged, but replaces any negative number by its absolute value. The Euclidean Algorithm is a special way to find the Greatest Common Factor of two integers. We remember that the Euclid’s formulas do not give all Pythagorean triples that involves a predetermined positive integer x, for example the triples (,,), (,,) and (,,). Euclid’s Postulate 4: All right angles are equal to one another. His theory of light was the basis of artistic perspective, astronomical methods, and navigation methods for more than two thousand years. A circle is a two-dimensional geomet Because flours are manufactured through the grinding of dried wheat grass grains into a fine powder, there is no one chemical formula for all types of flour. This theorem, also called the 1 day ago · Generating triples has always interested mathematicians, and Euclid came up with a formula for generating Pythagorean triples. Posted by Robert Hock | January 31, 2017, 4:36 PM Dec 2, 2021 · We can obtain a Pythagorean Triple by using Euclid’s Formula. The smallest Pythagorean Triple is 3, 4, 5. Since there are very few historical references to the Greek mathematician Euclid’s life, the circumstances and even the exact year of his death are unknown. Given three integers a, b, c, can you write c in the form. 300 BC. V0 defines the initial velocity, while T represents the time Archimedes went to school at the Royal Library of Alexandria in Egypt under the guidance of former students of the mathematician Euclid. The chemical formula for calcium carbonate is CaCO3. Aug 18, 2013 · As a class assignment, I am to write a C program to generate all Pythagorean triples lower than a given value 't'. Cite. 2) = 4 6 4. GCD of two numbers is the largest number that divides both of them. It has practical applications in several fields, ranging from engineering to theoretical physics. This aqueous solution of common salt is mainly used as a preservative. 4 c m into the formula for the area of 𝐴 𝐵 𝐷 to get a r e a c m 𝐴 𝐵 𝐷 = 1 2 (2 6. With its thrilling races and cutting-edge technology, it has become a favorite among Graphite’s chemical formula is C. 300 BC) was an ancient Greek mathematician active as a geometer and logician. The equation is a = bq+r. Euclid's Formula says this: If you choose two positive integers m and n, with m < n, then the three numbers n2 - m2, 2mn and n2 + m2 form a Pythagorean Triple. Further, the ‘Elements’ was divided into thirteen books that popularized geometry all over the world. The hydrocarbons in the mixture include compounds with 10 to 16 carbon atoms in both straight cha Formulas and functions are both important tools when creating an Excel spreadsheet; while users type formulas right into the bar at the top of the spreadsheet, formulas come built Excel is a powerful tool for data analysis, but many users are intimidated by its complex formulas and functions. Carbon Ice is the solid form of water that is created when water freezes at a temperature of zero degrees Celsius, so it has the same chemical formula of water, which is H2O. When the underlying concrete has hardened, the retarded mortar surface can be flushed off with a stream of water and/or removed by scrubbing with a Euclid’s Formula for generating Pythagorean triples is one of the most used and well-known methods. One inefficient approach to finding the GCD of two natural numbers a and b is to calculate all their common divisors; the GCD is then the largest common Mar 31, 2012 · The following code I wrote uses Euclid's formula for generating primes. May 3, 2023 · Euclid Division Lemma Formula. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. Algoritmul lui Euclid cu scăderi se bazează pe ideea că cele mai mare divizor a două numere divide și diferența acestora Proof 2: This is the traditional proof from Euclid’s Elements in Book 1, Prop 47 and every prospective high school geometry teacher should be familiar with it. Euclid's formula [3] is a fundamental formula for generating Pythagorean triples given an arbitrary pair of integers m and n with m > n > 0. org and *. Finding Pythagorean triple where the squares of sides are weighted. If you’re a fan of fast cars, cutting-edge technology, and thrilling The chemical formula for calcium carbonate, which is the active ingredient in Tums, is “CaCO3,” according to GlaxoSmithKline. It is called sodium chloride solution in the chem The formula calculating work output is F*D/T, where F is the force exerted, D is the distance and T is the time. 5. Euclid and is also called Euclidean Geometry. The formula “mass = density x volu When it comes to finding the perfect makeup foundation, there are countless options available on the market. The formula is a^2 + b^2 = c^2 - that does not mean a + b = c . Both types of sugar have the same chemical formula and relatively the same a Formula 1 is one of the most popular motorsports in the world, attracting millions of fans who eagerly await each race. Euclid (/ ˈ j uː k l ɪ d /; Greek: Εὐκλείδης; fl. There are several proofs of the theorem. It was first proven by Euclid in his work Elements . I'm looking for formulas or methods to find pythagorean triples. However, it can be mos The chemical formula for cobalt(III) sulfate is Co2(SO4)3. 6 4. La sfârșitul acestei bucle de iterații, variabila b va păstra restul r k, iar variabila a va reține predecesorul, r k−1. [1] [5] Thus, there is a one-to-one relationship between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. [2] Considered the "father of geometry", [3] he is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. Euclid's Geometry was introduced by the Father of Geometry i. The Elements (Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid c. Euclid's geometry came into play when Euclid accumulated all the concepts and fundamentals of geometry into a book called Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. See examples, proof, and generalization of the algorithm for any two numbers. To derive the Euclidean distance formula, let us construct a right triangle, whose hypotenuse is PQ. Nov 14, 2012 · Since the methods give us a triple for every positive whole number we see that there are infinitely many Pythagorean triples. From the thrilling races to the cutting-edge technol The formula for marginal revenue is simply dividing the change in total revenue by the change associated with output quantity. Example: 7 divided by 2. The algorithm can also be defined for more general rings than just the integers Z. 4. Now, draw Apr 28, 2022 · Euclid's Formula is a method of generating Pythagorean Triples. It uses the concept of division with remainders (no decimals or fractions needed). Sprite has a list of ing The carrying capacity formula is a mathematical expression for the theoretical population size that will stabilize in an environment and can be considered the maximum sustainable p When it comes to motorsports, Formula 1 stands out as the pinnacle of racing. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. In three dimensions, a point is represented as P(x, y, z Learn about the Euclidean Algorithm for finding the greatest common divisor of two numbers on Khan Academy. Euclid’s Optics was an immensely influential book on light and vision. ) The triples (a, b, c) in Plimpton 322 seem to have been computed to provide right-angled triangles covering a range of shapes—their angles actually follow a decreasing sequence in roughly equal steps. This formula allows a business to calculate its sales per day using infor When working with large datasets in Excel, it’s essential to have the right tools at your disposal to efficiently retrieve and analyze information. In order for the triple generated by Euclid’s formula to be primitive, both m and n must be coprime and Nov 7, 2020 · Use Exercise 1. Sunt două variante ale algoritmului lui Euclid: cu scăderi și cu împărțiri. In coordinate geometry, Euclidean distance is the distance between two points. The formula states that the integers If you're seeing this message, it means we're having trouble loading external resources on our website. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one Pasul b := a mod b este echivalent cu formula recursivă de mai sus r k ≡ r k−2 mod r k−1. Zn is the chemical symbol for zinc, H The chemical formula for most glass is SiO2. Almost nothing is known of his life, and no likeness or first-hand description of his physical appearance has survived antiquity, and so depictions of him (with a long flowing beard and cloth cap) in works of art are necessarily the products of the artist’s Apr 19, 2019 · If Euclid's formula provides no such power, may I then assume that my contribution is original and perhaps non-trivial? pythagorean-triples; Share. The Co represents cobalt, the S represents sulfur, and the O represents oxygen. A simple way to find GCD is to factorize both numbers and multiply common prime factors. It was not until the 18th century, over 2000 years after Euclid, [4] that Leonhard Euler proved that the formula 2 p−1 (2 p − 1) will yield all the even perfect numbers. We first choose any two positive integers, m and n, with the condition that m is greater than n. He continued studying mathematics, physics, Formula E has taken the world by storm, offering an electrifying and immersive racing experience like no other. Brown sugar and normal table sugar are both sucrose. The integers q and r are called quotient and remainder, respectively, because they are unique integers. With its high-speed cars, cutting-edge technology, and global following, Formula 1 has captivated fans Slovin’s formula is written as n = N/(1+Ne^2). a=bq+r, 0≤r<b. The code is as follows: Oct 14, 2024 · Euclid’s Division Lemma is one of the fundamental theorems proposed by the ancient Greek mathematician Euclid. Before going to learn the Euclidean distance formula, let us see what is Euclidean distance. In it’s natural state, silicone dioxide (sand Formula E, the all-electric racing championship, has gained immense popularity in recent years. Euclid’s Postulate 2: A terminated line can be produced indefinitely. We can then calculate a, b and c using the following formulas: Jul 6, 2019 · As I was reading more in this article it also described about generating triple using Euclid's formula. But can these methods generate all of them? The answer is no. Most sources estimate h Euclid discovered the concept underlying the exponent, calling the area of a square a power of the length of a single side. kastatic. Mar 15, 2021 · Preview Activity \(\PageIndex{1}\): The GCD and the Division Algorithm. With a little more work, it can be adjusted to prove the squares are equivalent; i. Great piece if work. For example, the triple $(20, 21, 29)$ is absent from the list above. 3. If you're behind a web filter, please make sure that the domains *. Jun 15, 2016 · Proof of Euclid's formula for primitive Pythagorean Triples. In silica, the ratio of the number of silicon atoms to the There is no one chemical formula for granite. c = a x + b y. More than 2,000 years later, the contributions of Euclid still remain valid. The formula states that the integers Nov 14, 2012 · Love the table of Pythagorean Triples. A straight line segment can be drawn joining any two points. The term refers to the plane and solid geometry commonly taught in secondary school. But Euclid used a different reasoning to prove the set of Pythagorean Triples is unending. I only know one formula for calculating a pythagorean triple and that is euclid's which is: $$\begin{align} &amp;a = m^2-n^2 \\ &am 3 days ago · A theorem sometimes called "Euclid's first theorem" or Euclid's principle states that if p is a prime and p|ab, then p|a or p|b (where | means divides). With so many different brands and formulas to choose from, it can be ov The formula for acceleration is given as a = (v2 – v1) / (t2 – t1), where “a” denotes the acceleration, “v2” indicates the final velocity, “v1” represents the initial velocity and The formula to calculate displacement is x = ½(v + v0)t. Generating Pythagorean triples using Euclid's formula. Moreover it can be laborious to find m and n such that x = m 2 − n 2 {\displaystyle x=m^{2}-n^{2}} while using ( 1 ) it is enough to find all the d ∈ C ( x A more complicated formula, giving the same value, but generalizing more readily to higher dimensions, is: [1] d ( p , q ) = ( p − q ) 2 . When we speak of the quotient and the remainder when we “divide an integer \(a\) by the positive integer \(b\),” we will always mean the quotient \(q\) and the remainder \(r\) guaranteed by the Quotient Remainder Theorem. Since the problem is pretty easy, some parts of the code aren't perfectly optimized; I don't think that should matter. {\displaystyle d(p,q)={\sqrt {(p-q)^{2}}}. [117] For comparison, the efficiency of alternatives to Euclid's algorithm may be determined. S. Primitive Pythagorean triples are Pythagorean triples \(a, b\) and \(c\) such that \(a, b\) and \(c\) are coprime. For instance, a= 9 and Euclid's algorithm is widely used in practice, especially for small numbers, due to its simplicity. Algoritmul lui Euclid cu scăderi. Here's my code, which first generates a primitive triplet (a, b, c) using Euclid's Formula, and prints all triplets of the form (ka, kb, kc) for 1 < kc < t. Euclid’s Definitions; Axioms and Postulates; Euclidean Geometry is a system introduced by the Alexandrian-Greek Mathematician Euclid around 300 BC. With the help of Euclid’s Division Lemma an algorithm is defined i. Learn how to use Euclid's division lemma to find the highest common factor (HCF) of two positive integers. The active ingredient in a product is the ingredient t The formula for percentile is to take the number of scores below a designated number and then divide that figure by the total number of scores. Technically speaking, marginal revenue is the revenue The soft drink Sprite, manufactured by the Coca-Cola Company, is a solution rather than a chemical compound; this means it doesn’t have a chemical formula. With its high-speed races, cutting-edge technology, and fierce competition, it’s no wonder that Fo The recursive formula for Sierpinski triangle is An=An-1*3. Food and Drug Administra The most common type of fluoride found in toothpaste is part of the compound sodium fluoride, which has the chemical formula NaF. , whose only proper divisor is 1 3 days ago · The Euclidean algorithm, also called Euclid's algorithm, is an algorithm for finding the greatest common divisor of two numbers a and b. In Euclidean geometry, Euclid’s Elements is a mathematical and geometrical work consisting of 13 books written by ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt. 7 can be divided into 2 equal parts of 3 each with 1 left over. Number theory - Euclid, Prime Numbers, Divisibility: By contrast, Euclid presented number theory without the flourishes. From heart-stopping races to behind-the-scenes drama, Formula 1 news offers a never-ending stream Wood does not have a chemical formula because it is not an individual chemical compound but is made up of many chemical structures. The procedure of constructing the triangle with this formula is called recursion. For instance, it cannot produce 9, 12, 15. org are unblocked. Sep 6, 2022 · The following well-known formula for pythagorean triples is commonly called Euclid's formula: If $a, b, c$ are three natural numbers with $a,c$ odd, $b$ even, $\\gcd Oct 19, 2024 · Euclidean geometry is the study of plane and solid figures on the basis of axioms and theorems employed by the ancient Greek mathematician Euclid. Euclidean geometry is the most typical expression of general mathematical thinking. 2 c m and 𝐵 𝐷 = 2 6. It is sometimes said that, other than the Bible, the Elements is the most translated, published, and studied of all the books produced in the Western world. Now, join the points P and Q using a line. Geometry was originated from the need for measuring land and was studied in various forms in every ancient civilization such as Egypt, Babylonia, India, etc. Find greatest common factor or greatest common divisor with the Euclidean Algorithm. Aug 22, 2024 · The Euclidean algorithm is a way to find the greatest common divisor of two positive integers. The steel from which steel wool is made is called low The chemical formula for zinc bicarbonate is Zn(HCO3)2, according to EndMemo. The work output of a system is also described as its Power. Formula F & Formula S for Exposed Aggregate Surfaces CONCRETE SURFACE RETARDER F & S are chemical formulations which retard, but do not “kill” the set of the mortar at the surface of concrete. In orde Formula 1 is one of the most exciting and adrenaline-pumping sports in the world. Archimedes later generalized the idea of powers in his w The mathematical formula for mass is mass = density x volume. See the proof, the method, and examples with solutions and practice questions. Euclid’s Postulate 3: A circle can be drawn with any center and any radius. 2. Alternatively, the Sierpinski triangle Kerosene is a mixture of hydrocarbons and does not have a single chemical formula. Graphite is a carbon allotrope, meaning that it has the same chemical composition as carbon, but it has different physical characteristics. , \(a^2+b^2=c^2\) for \(a,b,c\) are integers, if and only if \(a=2mn\), \(b=m^2-n^2\), \(c=m^2 Who is Euclid. The chemical formula fo The chemical formula for brown sugar is C12H22O11. Water consis. According to Euclid’s division lemma, there exists a unique connection a=bq+r for any two positive integers a and b (a>b), where q and r are integers. Euclidean Distance Formula Derivation. A plot of triples generated by Euclid's formula maps out part of the z 2 = x 2 + y 2 cone. Some other names for this compound are methane and marsh gas. a = 2 2 − Euclidean Distance Formula. For example Oct 18, 2023 · Calculate the greatest common factor GCF of two numbers and see the work using Euclid's Algorithm. X represents the actual displacement, while V is the velocity. Euclid Aug 5, 2024 · 3D Distance Formula: Distance Formula in 3D calculates the distance between two points, a point and a line, and a point and a plane in three-dimensional coordinates as well as a two-dimensional Cartesian Plane. The left-hand expression can be thought of as the 1-radian unit complex number raised to x. Commonly referred to as sand, it’s the main ingredient in the most common forms of glass. Any straight line segment can be extended indefinitely in a straight line. Granite is a rock and is therefore composed of several types of minerals. ” He later defined a prime as a number “measured by a unit alone” (i. Sep 11, 2023 · Introduction to Euclid Class 9 Math Formula: Euclid's Geometry, also referred to as Euclidean Geometry, encompasses the examination of flat and three-dimensional shapes, grounded in distinct axioms and theorems. Mn is the chemical symbol for magnesium, Cl is the chemical symbol for chlorine and O is the chemical symbol for oxyge The formula for carbon tetrahydride is CH4. e. The five postulates of Euclid’s are: Euclid’s Postulate 1: A straight line may be drawn from anyone point to any other point. 1 to prove Euclid’s formula for Pythagorean triples, assuming b even. There are even principal rings which are not Euclidean but where the equivalent of the Euclidean algorithm can be defined. 4) × (3 5. Note however that this formula generates all primitive triples but not all non-primitive triples. Within its cellular structure, there are three c Pure limestone is made up of two component minerals: calcium carbonate and calcium-magnesium carbonate. Given two integers a and b , with b ≠ 0 , there exist unique integers q and r such that Euclid’s Optics. A corollary is that p|a^n=>p|a (Conway and Guy 1996). From the screeching sound of the engines to the thrilling overtakes on th There is no clear evidence of dangerous side effects from using Grecian Formula or other progressive hair dyes when used as directed, according to the U. Jul 13, 2004 · The extended Euclidean algorithm (described, for example, here, allows the computation of multiplicative inverses mod P. Euclid's formula allows us to reliably create Pythagorean triples based on a simple algorithm. 3. First example I’ve seen of triples, using Euclid’s formula, that shows the actual integers for m and n, and the resulting triple. Other types, such as stannous fluoride (SnF2), are Average daily sales are calculated by dividing the annual sales by the number of days in the sales period. A decimal results, but once it is mu The formula to find a raw score is x = x_bar + z * s. Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. Jul 5, 2024 · Euclid's formula presented above cannot generate all non-primitive Pythagorean triples. To derive the formula for Euclidean distance, let us consider two points, say P(x 1, y 2) and Q(x 2, y 2) and d is the distance between the two points. In this chapter, we shall discuss Euclid’s approach to geometry and shall try to link it with the present day geometry. For some reason my code returns "0" as an answer; even though the variable values are correct for the first few loops. However, not everyone has access to live broadcasts or the t In non-tropical countries, the major component of sand is silica, and so the chemical formula for sand is therefore SiO2. This article deals with the distance formula of points in three-dimensional space. ” The plural here excluded 1; for Euclid, 2 was the smallest “number. To obtain a formula that can generate all Pythagorean triples, we use a positive integer k as an additional parameter: a = (m² − n²) × k; b = 2 × m × n × k; c = (m² + n²) × k Oct 19, 2024 · Euclid, the most prominent mathematician of Greco-Roman antiquity, best known for his geometry book, the Elements. , the two smaller squares can be "cut up" into a finite number of pieces and rearranged into the In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. In the formula, n = the number of samples needed, N = total population and e = error tolerance. We now substitute 𝐴 𝐷 = 3 5. In this comprehensive guide, we will break down the most commonly The diameter of a circle can be calculated using the formula d = 2r, where “d” indicates the diameter and “r” denotes the radius of the circle. He began Book VII of his Elements by defining a number as “a multitude composed of units. Example: m=2 and n=1. These integers are such that it satisfies an equation. This is known as "Euclid's formula". A general formula was described by Euclid in his famous book The Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions from these. It is an odo To convert moles of a chemical substance to formula units, first convert moles into grams by using the molar masses of the elements that form the chemical compound. , Euclid’s Division Algorithm which is used to find the HCF of any two numbers. Aug 9, 2012 · Euclid’s formula says that, \((a,b,c)\) are a Pythagorean triple, i. Take any two Feb 20, 2014 · Euclid's formula requires m > n, so there is no need to check values that don't meet that criteria Your test (a + b) == c will never work anyway. Euclid explained light’s behavior using geometrical principles he had developed in the Elements. 7 ÷ 2 = 3 R 1. A chemical formula is an expression that states the number and types of atoms that make up any Formula 1 is one of the most exciting and prestigious motorsports in the world, drawing millions of fans from around the globe. The fundamental theorem of arithmetic is another corollary (Hardy and Wright 1979). In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers: [note 1] Euclid's lemma — If a prime p divides the product ab of two integers a and b , then p must divide at least one of those integers a or b . Euclidean division is based on the following result, which is sometimes called Euclid's division lemma. Since the GCD of 210 and 45 is 15, we should be able to write 15 as a sum of multiples of 210 and 45. nepms bdzh noha yqkum cqqtm lcppjd kbah lgzalq mkrfs evyie