Tiger Algebra/core — French @ Weblate: The least common multiple (LCM), sometimes called the lowest common …

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The least common multiple (LCM), sometimes called the lowest common multiple or least common divisor (LCD), is the smallest number that two or more selected numbers can divide evenly into without a remainder. For example, <math>21</math> is the LCM of <math>3</math> and <math>7</math> because it is the lowest number that they both divide evenly into. 

There are several ways to find the LCM: listing the multiples of each number, prime factorization, the cake or ladder method, the division method, or using the greatest common factor. 

The Tiger Algebra LCM calculator not only finds the LCM but also shows the individual steps involved in doing so, helping you better understand and remember the process!

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Key	English	French
FactoringPolynomialsHavingMoreThan3Terms	Factoring polynomials with four or more terms	Factorisation de polynômes de quatre termes ou plus
FactoringPolynomialsHavingMoreThan3TermsContent	A simple way to approach factoring a polynomial with four or more terms is by grouping it into sets of two. This method involves examining these sets together to see whether a particular technique could be applied to both. One technique to start with is checking whether it is possible to find the greatest common factor (GCF) between a set of two terms. If the GCF cannot be found, the polynomials could be grouped in another way and examined for another technique. There is always the possibility that the polynomial is prime and cannot be factored. <div style="white-space: normal;"> <p>Factoring polynomials with four or more terms can be more complex than factoring binomials or trinomials. However, there are strategies that can help simplify the process.</p> <h2>Step-by-Step Process</h2> <p>To factor polynomials with four or more terms, follow these steps:</p> <ol> <li>Group the terms in pairs.</li> <li>Factor each pair using common factoring techniques such as GCF, difference of squares, or trinomial factoring.</li> <li>Look for a common factor among the resulting expressions.</li> <li>Factor out the common factor.</li> <li>Express the polynomial as the product of the common factor and the remaining expressions.</li> <li>Check your work by multiplying the factors to ensure you get the original polynomial.</li> </ol> <h2>Example</h2> <p>Let's factor the polynomial <math>x^3 + 2x^2 - 3x - 6</math>:</p> <p>Step 1: Group the terms - <math>(x^3 + 2x^2) - (3x + 6)</math>.</p> <p>Step 2: Factor each pair - <math>x^2(x + 2) - 3(x + 2)</math>.</p> <p>Step 3: Look for a common factor - Both expressions share <math>(x + 2)</math> as a common factor.</p> <p>Step 4: Factor out the common factor - <math>(x + 2)(x^2 - 3)</math>.</p> <p>Step 5: The polynomial is now factored as <math>(x + 2)(x^2 - 3)</math>.</p> <p>Factoring polynomials with four or more terms requires practice and patience, but mastering this skill can greatly simplify algebraic expressions and equations.</p> </div>	Une façon simple d’aborder la factorisation d’un polynôme de quatre termes ou plus est de le regrouper en ensembles de deux. Cette méthode consiste à examiner ces ensembles pour voir si une technique particulière pourrait être appliquée aux deux. Une première technique consiste à vérifier s’il est possible de trouver le plus grand commun diviseur (PGCD) entre un ensemble de deux termes. Si le PGCD ne peut pas être trouvé, les polynômes pourraient être regroupés d’une autre manière et examinés pour trouver une autre technique. Il y a toujours la possibilité que le polynôme soit premier et ne puisse pas être factorisé.
FactorTrinomials	Factor trinomials	Factorisation de trinômes
FactorTrinomialsContent	<div style="white-space: normal;"> <p>Factoring trinomials is an essential skill in algebra, particularly in polynomial factorization. Trinomials are algebraic expressions with three terms, and factoring involves breaking them down into the product of two or more simpler expressions.</p> <h2>Basic Concepts</h2> <p>To factor trinomials, it's crucial to understand the following concepts:</p> <ul> <li><strong>Trinomials:</strong> Algebraic expressions with three terms, typically in the form <math>ax^2 + bx + c</math>.</li> <li><strong>Factoring:</strong> The process of expressing an algebraic expression as the product of its factors.</li> <li><strong>Factorization techniques:</strong> Methods used to factor trinomials, such as trial and error, grouping, and the quadratic formula.</li> </ul> <h2>Factoring Techniques</h2> <p>There are various techniques to factor trinomials:</p> <ul> <li><strong>Trial and error:</strong> Trying different combinations of factors until finding the correct one.</li> <li><strong>Grouping:</strong> Grouping the terms of the trinomial and factoring common factors.</li> <li><strong>Quadratic formula:</strong> Applying the quadratic formula to find the roots of the trinomial.</li> </ul> <h2>Examples</h2> <p>Let's consider a few examples to illustrate the factoring of trinomials:</p> <h3>Example 1:</h3> <p>Factor <math>x^2 + 5x + 6</math></p> <p>We look for two numbers that multiply to give 6 and add to give 5. These numbers are 2 and 3. So, <math>x^2 + 5x + 6 = (x + 2)(x + 3)</math></p> <h3>Example 2:</h3> <p>Factor <math>2x^2 + 7x + 3</math></p> <p>Using the quadratic formula, we find the roots to be <math>x = -\frac{1}{2}</math> and <math>x = -3</math>. So, <math>2x^2 + 7x + 3 = 2(x + \frac{1}{2})(x + 3)</math></p> <h2>Conclusion</h2> <p>Factoring trinomials is an important skill in algebra that finds applications in various mathematical problems and real-world scenarios. Mastering the techniques of trinomial factoring enhances problem-solving abilities and deepens understanding of algebraic concepts.</p> </div>	Tiger Algebra te montre, étape par étape, comment factoriser des trinômes selon différentes méthodes.
FindAreaofTrianglegivenbyits3sides	Find area of triangle given by its 3 sides	Détermination de l’aire d’un triangle à partir de ses 3 côtés
FindAreaofTrianglegivenbyits3sidesContent	To find the Area of Triangle given its 3 sides, enter the sizes of the three sides seperated by commas and wrapped in parentheses. Tiger Algebra will show you the step by step solution. Sample input: <math>(3,4,5)</math> <div style="white-space: normal;"> <p>To find the area of a triangle given its three sides (also known as the side-length formula), you can use Heron's formula. Heron's formula allows you to calculate the area of a triangle using its side lengths alone.</p> <h2>Heron's Formula</h2> <p>Heron's formula states that the area <math>A</math> of a triangle with side lengths <math>a</math>, <math>b</math>, and <math>c</math> is given by:</p> <div style="margin-left: 20px;"> <math>A = \sqrt{s(s - a)(s - b)(s - c)}</math>,<br> <i>where</i> <math>s</math> is the semi-perimeter of the triangle, calculated as:<br> <math>s = \frac{a + b + c}{2}</math>. </div> <h2>Example</h2> <p>Let's find the area of a triangle with side lengths 5, 6, and 7:</p> <div style="margin-left: 20px;"> <math>a = 5, b = 6, c = 7</math><br> <math>s = \frac{5 + 6 + 7}{2} = 9</math><br> <math>A = \sqrt{9(9 - 5)(9 - 6)(9 - 7)}</math><br> <math>A = \sqrt{9(4)(3)(2)} = \sqrt{216}</math><br> <math>A \approx 14.7</math>. </div> <p>Therefore, the area of the triangle is approximately 14.7 square units.</p> <p>Heron's formula provides an efficient method for finding the area of a triangle when you know the lengths of its sides.</p> </div>	Pour trouver l’aire d’un triangle à partir de ses 3 côtés, entrer les dimensions des trois côtés, séparées par des virgules et mises entre parenthèses. Tiger Algebra te montrera la solution étape par étape. Exemple : <math>(3,4,5)</math>
FindPrimeFactors	Find prime factors	Détermination de facteurs premiers
FindPrimeFactorsContent	<div style="white-space: normal;"> <p>Prime factorization is the process of finding the prime numbers that multiply together to give a specific number. This is often used in mathematics to simplify fractions or solve problems involving factors.</p> <h2>Step 1: Begin with the Number</h2> <p>To find the prime factors of a number, start with the number itself. Let's take an example: 72.</p> <h2>Step 2: Divide by Prime Numbers</h2> <p>Start dividing the number by the smallest prime number, 2, and continue dividing by prime numbers until you cannot divide evenly anymore.</p> <p>For example, for 72:</p> <ul> <li>72 ÷ 2 = 36</li> <li>36 ÷ 2 = 18</li> <li>18 ÷ 2 = 9</li> <li>9 ÷ 3 = 3</li> <li>3 is already a prime number.</li> </ul> <h2>Step 3: Record the Prime Factors</h2> <p>The prime factors of 72 are 2, 2, 2, 3, and 3.</p> <h2>Step 4: Check Your Work</h2> <p>Multiply all the prime factors together to ensure you get the original number: 2 * 2 * 2 * 3 * 3 = 72.</p> <p>Finding prime factors is essential for many mathematical operations, including simplifying fractions, finding greatest common divisors, and more.</p> </div>	Un nombre premier ne peut être divisé que par un seul nombre ou par lui-même. Les nombres entiers composites (non premiers) peuvent être écrits comme un produit de leurs facteurs premiers. Entre des facteurs (n’importe quel nombre) ou des nombres premiers (n’importe quel nombre) dans la calculatrice d’algèbre et Tiger trouvera ses facteurs premiers pour toi.
GCFCalculator	Greatest common factor	Plus grand facteur commun
GCFCalculatorContent	The greatest common factor (GCF), sometimes also referred to as the highest common factor (HCF) or the greatest common divisor (GCD), is the largest positive integer that a set of integers can all be divided by. For example, the largest number that 12, 24, and 32 can all be divided by is 4, so their greatest common factor is 4. Similarly, the largest number that 3, 5, and 10 can all be divided by is 1, so their greatest common factor is 1. <br><br> There are two ways to find the greatest common factor: listing the factors of each number and prime factorization. <br><br> <b>Method 1: Listing the factors of each number</b><br> List all of the factors of each number and identify the largest (greatest) factor that all the numbers have in common. <br><br> 12 - 1, 2, 3, <b>4</b>, 6, 12<br> 24 - 1, 2, 3, <b>4</b>, 6, 8, 12, 24<br> 32 - 1, 2, 3, <b>4</b>, 8, 16, 32<br><br> 4 is the greatest factor the numbers have in common, so it is the greatest common factor. <br><br> <b>Method 2: Prime factorization</b><br> Use a factor tree to identify the prime factors (factors that are prime numbers) of each number. Identify the prime factors that all the numbers have in common and multiply them together to get the greatest common factor. <br> <img src="/images/gcfTree1.png" alt="Prime factorization tree"><br> The prime factors that all of the numbers have in common are 2 and 2. Multiply these together to get the greatest common factor, 4.	Le plus grand commun diviseur (PGCD), parfois également appelé plus grand facteur commun (PFC) ou plus grand diviseur commun (GDC), est le plus grand entier positif par lequel un ensemble d'entiers peut tous être divisé. Par exemple, le plus grand nombre que 12, 24 et 32 peuvent tous être divisés par est 4, donc leur plus grand facteur commun est 4. De même, le plus grand nombre que 3, 5 et 10 peuvent tous être divisés est de 1, donc leur plus grand facteur commun est de 1. <br><br> Il y a deux manières de trouver le plus grand commun diviseur : énumérer les facteurs de chaque nombre et la factorisation en nombres premiers. <br><br> <b>Méthode 1 : Énumération des facteurs de chaque nombre</b><br> Énumérez tous les facteurs de chaque nombre et identifiez le plus grand (greatest) facteur que tous les nombres ont en commun. <br><br> 12 - 1, 2, 3, <b>4</b>, 6, 12<br> 24 - 1, 2, 3, <b>4</b>, 6, 8, 12, 24<br> 32 - 1, 2, 3, <b>4</b>, 8, 16, 32<br><br> 4 est le plus grand facteur que les nombres ont en commun, donc c'est le plus grand commun diviseur. <br><br> <b>Méthode 2 : Factorisation en nombres premiers</b><br> Utilisez un arbre de facteurs pour identifier les facteurs premiers (facteurs qui sont des nombres premiers) de chaque nombre. Identifiez les facteurs premiers que tous les nombres ont en commun et multipliez-les ensemble pour obtenir le plus grand commun diviseur. <br> <img src="/images/gcfTree1.png" alt="Arbre de factorisation prime"><br> Les facteurs premiers que tous les nombres ont en commun sont 2 et 2. Multipliez ces derniers ensemble pour obtenir le plus grand commun diviseur, 4.
GeometricSeries	Geometric series	Série géométrique
GeometricSeriesContent	In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series <math>1,2,4,8</math> is geometric, because each successive term can be obtained by multiplying the previous term by <math>2</math>. <div style="white-space: normal;"> <p>A geometric series is the sum of the terms of a geometric sequence. It is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.</p> <h2>General Form</h2> <p>The general form of a geometric series is:</p> <div style="margin-left: 20px;"> <math>a + ar + ar^2 + ar^3 + \ldots = \sum_{n=0}^{\infty} ar^n</math>, </div> <p>where <math>a</math> is the first term and <math>r</math> is the common ratio.</p> <h2>Sum Formula</h2> <p>The sum of a finite geometric series with <math>n</math> terms is given by the formula:</p> <div style="margin-left: 20px;"> <math>S_n = a \frac{1 - r^n}{1 - r}</math>, </div> <p>where <math>S_n</math> is the sum of the first <math>n</math> terms.</p> <h2>Properties</h2> <ul> <li>If the absolute value of the common ratio <math>r</math> is less than 1, the series converges to a finite value.</li> <li>If the absolute value of <math>r</math> is greater than or equal to 1, the series diverges.</li> <li>The sum of an infinite geometric series can be found using the formula for the sum of an infinite geometric series:</li> <div style="margin-left: 20px;"> <math>S_{\infty} = \frac{a}{1 - r}</math>. </div> </ul> <h2>Applications</h2> <p>Geometric series have various applications in mathematics, physics, engineering, and finance. They are used to model growth and decay processes, calculate interest, analyze circuits, and more.</p> <p>Understanding geometric series and their properties is essential for solving problems in many fields.</p> </div>	En mathématiques, une série géométrique est une série avec un rapport constant entre les termes successifs. Par exemple, la série <math>1,2,4,8</math> est géométrique car chaque terme successif peut être obtenu en multipliant le terme précédent par <math>2</math>.
IdentifyingPerfectCubes	Identifying perfect cubes	Identification de cubes parfaits
IdentifyingPerfectCubesContent	<div style="white-space: normal;"> <p>In general, a perfect cube is represented as <math>n^3</math>, where <math>n</math> is an integer.</p> <p>There are several techniques to identify perfect cubes:</p> <ul> <li><strong>Examine the last digit:</strong> Perfect cubes have specific patterns in their last digit. By examining the last digit of a number, you can often determine if it's a perfect cube.</li> <li><strong>Prime factorization:</strong> Another method is to perform prime factorization of the number and check if each prime factor appears in groups of three.</li> <li><strong>Use a table of cubes:</strong> Memorizing the cubes of numbers from 1 to 10 can help quickly identify perfect cubes.</li> </ul> <h2>Examples</h2> <p>Let's consider a few examples to illustrate the identification of perfect cubes:</p> <h3>Example 1:</h3> <p>Identify whether 64 is a perfect cube.</p> <p>Using the examination of the last digit method, we find that the last digit of 64 is 4. The last digits of perfect cubes cycle in a specific pattern (0, 1, 8, 7, 4, 5, 6, 3, 2, 9), and 4 is in this cycle. Hence, 64 is a perfect cube.</p> <h3>Example 2:</h3> <p>Identify whether 1000 is a perfect cube.</p> <p>By performing prime factorization, we find that <math>1000 = 2^3 \times 5^3</math>. Since both 2 and 5 appear in groups of three, 1000 is a perfect cube.</p> <h2>Conclusion</h2> <p>Identifying perfect cubes is an important skill in mathematics that finds applications in various fields. By understanding the basic concepts and techniques, one can easily recognize perfect cubes and utilize them in problem-solving.</p> </div>	Un cube parfait est le cube d’un autre polynôme. Le solveur Tiger Algebra trouve des cubes parfaits et te montre des solutions, étape par étape.
LCMCalculator	Least common multiple (LCM)	Calculatrice de PPCM
LCMCalculatorContent	The least common multiple (LCM), sometimes called the lowest common multiple or least common divisor (LCD), is the smallest number that two or more selected numbers can divide evenly into without a remainder. For example, <math>21</math> is the LCM of <math>3</math> and <math>7</math> because it is the lowest number that they both divide evenly into.<br><br> There are several ways to find the LCM: listing the multiples of each number, prime factorization, the cake or ladder method, the division method, or using the greatest common factor.<br><br> The Tiger Algebra LCM calculator not only finds the LCM but also shows the individual steps involved in doing so, helping you better understand and remember the process!	Le plus petit commun multiple (PPCM), parfois appelé le plus petit commun multiple ou le plus petit commun diviseur, est le plus petit nombre pouvant être divisé par deux ou plusieurs nombres sélectionnés sans reste. Par exemple, <math>21</math> est le PPCM de <math>3</math> et <math>7</math> car c'est le plus petit nombre pouvant être divisé par eux sans reste.<br><br> Il y a plusieurs façons de trouver le PPCM : en énumérant les multiples de chaque nombre, en factorisant les nombres premiers, en utilisant la méthode du cake ou de l'échelle, la méthode de la division ou en utilisant le plus grand facteur commun.<br><br> La calculatrice Tiger Algebra PPCM trouve non seulement le PPCM, mais montre les différentes étapes impliquées dans ce processus, t’aidant ainsi à mieux comprendre et te souvenir du processus !
LinearEquationswithOneUnknown	Linear equations with one unknown	Équations linéaires à une inconnue
LinearEquationswithOneUnknownContent	The main application of linear equations is solving problems in which an unknown variable, usually (but not always) x, is dependent on a known constant.<br>We solve linear equations by isolating the unknown variable on one side of the equation and simplifying the rest of the equation. When simplifying, <b>anything that is done to one side of the equation must also be done to the other</b>.<br><br>An equation of:<br> <math>ax+b=0</math><br> in which <math>a</math> and <math>b</math> are the constants and <math>x</math> is the unknown variable, is a typical linear equation with one unknown. To solve for <math>x</math> in this example, we would first isolate it by subtracting <math>b</math> from both sides of the equation. We would then divide both sides of the equation by <math>a</math>, resulting in an answer of:<br><math>x = -b/a</math>	La principale application des équations linéaires est la résolution de problèmes dans lesquels une variable inconnue, généralement (mais pas toujours) x, dépend d'une constante connue.<br>Nous résolvons les équations linéaires en isolant la variable inconnue d'un côté de l'équation et en simplifiant le reste de l'équation. Lors de la simplification, <b>tout ce qui est fait d'un côté de l'équation doit également être fait de l'autre côté</b>.<br><br>L’équation :<br> <math>ax+b=0</math><br> dans laquelle <math>a</math> et <math>b</math> sont des constantes et <math>x</math> est la variable inconnue, est une équation linéaire typique à une inconnue. Pour trouver <math>x</math> dans cet exemple, nous devons d'abord l'isoler en soustrayant <math>b</math> des deux côtés de l'équation. Nous divisons ensuite les deux côtés de l'équation par <math>a</math>, ce qui donne le résultat suivant :<br><math>x = -b/a</math>
LinearEquationswithThreeUnknowns	Linear equations with three unknowns	Équations linéaires à trois inconnues
LinearEquationswithThreeUnknownsContent	A linear system of three equations with three variables is any system that can be written in the form.<br><br> where the constants (a,b,c,d,e,f,g,h and i) can be zero as long as each equation has at least one variable (x,y or z) in it. <br><br> In addition, for the system to be called linear, the variables can only to the first power, are only in the numerator and there are no products of variables in any of the equations. <br><br> A solution to a system of equations are values of x, y and z such that, when substituted into the equations, satisfies both equations at the same time. <br><br> Separate your equations with a semi-colon ";"" when feeding them to the Tiger.	Un système linéaire de trois équations avec trois variables est tout système qui peut être écrit sous la forme.
LinearEquationswithTwoUnknowns	Linear equations with two unknowns	Équations linéaires à deux inconnues
LinearEquationswithTwoUnknownsContent	A linear system of two equations with two variables is any system that can be written in the form. <br><br> <math>ax+by=p</math> <br> <math>cx+dy=q</math> <br><br> where the constants (a,b,c and d) can be zero as long as each equation has at least one variable (x or y) in it. In addition, for the system to be called linear, the variables can only to the first power, are only in the numerator and there are no products of variables in any of the equations. <br><br> A solution to a system of equations is a value of x and a value of y that, when substituted into the equations, satisfies both equations at the same time. <br><br> Separate you equations with a semi-colon ";" when feeding them to the Tiger	Un système linéaire de deux équations avec deux variables est tout système qui peut être écrit sous la forme. <br><br> <math>ax+by=p</math> <br> <math>cx+dy=q</math> <br><br> où les constantes (a, b, c et d) peuvent être zéro tant que chaque équation contient au moins une variable (x ou y). De plus, pour que le système soit dit linéaire, les variables ne peuvent être qu'à la première puissance, ne se trouvent que dans le numérateur et il n'y a aucun produit de variables dans les équations. <br><br> La solution d'un système d'équations est constituée des valeurs x, y et z qui, lorsqu'elles sont substituées dans les équations, répondent aux deux équations en même temps. <br><br> Sépare tes équations par un point-virgule « ; » lorsque tu les saisis dans Tiger
LinearEquationsWithFourUnknowns	Linear equations with four unknowns	Équations linéaires à quatre inconnues
LinearEquationsWithFourUnknownsContent	A group of four linear equations with four unknown factors forms a system of equations. To solve this system means finding the value of the unknown factors in a manner that verifies all of the equations in the system. The overall concept behind solving a system of equations is combining the equations in such a way that the number of variables is reduced. This can be done by substitution or elimination (also called row reduction), but also by graphing or using matrices. <div style="white-space: normal;"> <p>Linear equations with four unknowns are equations where each term is either a constant or a product of a constant and one of the four variables raised to the power of 1. The general form of such equations is:</p> <div style="margin-left: 20px;"> <math>a_1x + b_1y + c_1z + d_1w = k_1</math><br> <math>a_2x + b_2y + c_2z + d_2w = k_2</math><br> <math>a_3x + b_3y + c_3z + d_3w = k_3</math><br> <math>a_4x + b_4y + c_4z + d_4w = k_4</math>, </div> <p>where <math>x</math>, <math>y</math>, <math>z</math>, and <math>w</math> are the unknown variables, and <math>a_i, b_i, c_i, d_i</math> (for <math>i = 1, 2, 3, 4</math>) and <math>k_i</math> are constants.</p> <h2>Solving Methods</h2> <p>There are several methods to solve systems of linear equations with four unknowns, including:</p> <ul> <li><strong>Matrix Methods:</strong> Such as Gauss elimination or Cramer's rule.</li> <li><strong>Substitution:</strong> Solve one equation for one variable and substitute it into the other equations.</li> <li><strong>Elimination:</strong> Add or subtract equations to eliminate one variable at a time.</li> <li><strong>Row Reduction:</strong> Use row reduction techniques to transform the augmented matrix to row-echelon or reduced row-echelon form.</li> </ul> <h2>Example</h2> <p>Let's consider the following system of linear equations with four unknowns:</p> <div style="margin-left: 20px;"> <math>3x + 2y - z + 4w = 7</math><br> <math>2x - y + 3z - 2w = -5</math><br> <math>x + 2y + 2z - 3w = 8</math><br> <math>4x - y - z + 2w = -3</math>.<br> </div> <p>We can solve this system using any of the methods mentioned above to find the values of <math>x</math>, <math>y</math>, <math>z</math>, and <math>w</math>.</p> <p>Understanding how to solve systems of linear equations with four unknowns is essential for various applications in mathematics, physics, engineering, and other fields.</p> </div>	Un groupe de quatre équations linéaires avec quatre facteurs inconnus forme un système d’équations. Pour résoudre ce système, il s’agit de trouver la valeur des facteurs inconnus qui vérifie toutes les équations du système. Le concept global derrière la résolution d’un système d’équations est la combinaison d’équations de telle sorte que le nombre de variables est réduit. Cela peut se faire par substitution ou élimination (aussi appelée réduction des lignes), mais aussi en établissant un graphique ou en utilisant des matrices.
OtherFactorizations	Other factorizations	Autres factorisations
OtherFactorizationsContent	<div style="white-space: normal;"> <p>Factorization involves breaking down a number or algebraic expression into its constituent factors. Besides the common factorization techniques like prime factorization and polynomial factorization, there are other specialized methods used in specific contexts.</p> <h2>Specialized Techniques</h2> <ul> <li><strong>Difference of squares:</strong> Factoring expressions of the form \(a^2 - b^2\).</li> <li><strong>Sum or difference of cubes:</strong> Factoring expressions of the form \(a^3 + b^3\) or \(a^3 - b^3\).</li> <li><strong>Quadratic trinomials:</strong> Factoring quadratic expressions of the form \(ax^2 + bx + c\).</li> </ul> <h2>Applications</h2> <p>These specialized factorization techniques are commonly used in algebraic manipulations, simplifications, and solving equations.</p> </div>	<div style="white-space: normal;"> <p>La factorisation consiste à décomposer un nombre ou une expression algébrique en ses facteurs constitutifs. Outre les techniques de factorisation courantes comme la factorisation par les nombres premiers et la factorisation polynomiale, il existe d'autres méthodes spécialisées utilisées dans des contextes spécifiques.</p> <h2>Techniques Spécialisées</h2> <ul> <li><strong>Différence des carrés :</strong> Factorisation des expressions de la forme \(a^2 - b^2\).</li> <li><strong>Somme ou différence de cubes :</strong> Factorisation des expressions de la forme \(a^3 + b^3\) ou \(a^3 - b^3\).</li> <li><strong>Trinômes quadratiques :</strong> Factorisation des expressions quadratiques de la forme \(ax^2 + bx + c\).</li> </ul> <h2>Applications</h2> <p>Ces techniques spécialisées de factorisation sont couramment utilisées dans les manipulations algébriques, les simplifications et la résolution d'équations.</p> </div>
SolvingLinearEquationsBySubstitution	Solving linear equations by substitution	Résolution d’équations linéaires par substitution
SolvingLinearEquationsBySubstitutionContent	Substitution is one of the methods for solving linear equations. It is a good choice if there is a variable with a coefficient of 1. This method involves solving one of the equations for a variable and then substituting that expression into another equation to solve for another variable. This will produce a single equation with one variable, which can then be solved algebraically. <div style="white-space: normal;"> <p>Solving linear equations by substitution is a method used to find the values of variables in an equation system. This method involves solving one of the equations for one variable and then substituting that expression into the other equations.</p> <h2>Basic Steps</h2> <p>The basic steps for solving linear equations by substitution are as follows:</p> <ol> <li><p>Choose one of the equations and solve it for one variable in terms of the other variables.</p></li> <li><p>Substitute the expression found in step 1 into the other equations.</p></li> <li><p>Solve the resulting equations for the remaining variables.</p></li> <li><p>Check the solutions obtained by substituting them back into the original equations to verify their correctness.</p></li> </ol> <h2>Example</h2> <p>Let's solve the following system of linear equations by substitution:</p> <div style="margin-left: 20px;"> <math>2x + y = 8</math><br> <math>x - 3y = -5</math>.<br> </div> <p>From the first equation, we can solve for <math>y</math>:</p> <div style="margin-left: 20px;"> <math>y = 8 - 2x</math>.<br> </div> <p>Now, we substitute this expression for <math>y</math> into the second equation:</p> <div style="margin-left: 20px;"> <math>x - 3(8 - 2x) = -5</math>.<br> </div> <p>Solving this equation gives us the value of <math>x</math>. Once we have the value of <math>x</math>, we can substitute it back into the expression we found for <math>y</math> to find the value of <math>y</math>.</p> <p>Solving linear equations by substitution is a useful technique for finding solutions to systems of equations when the equations are linear.</p> </div>	La substitution est l’une des méthodes de résolution d’équations linéaires. C’est un bon choix s’il y a une variable avec un coefficient de 1. Cette méthode consiste à résoudre l’une des équations d’une variable, puis à substituer cette expression à une autre équation à résoudre pour une autre variable. Ceci produira une équation simple avec une variable qui pourra alors être résolue algébriquement.
LinearInequalities	Linear inequalities	Inégalités linéaires
MultiplyingExponents	Multiplying exponents	Multiplication d’exposants
MultiplyingExponentsContent	Exponents are a shorter way of noting repeated multiplication of the same thing by itself. <br> For example <math>777*7=7^4</math> <br> The 'exponent', 4 in the above example, tells us how many times the value is being multiplied. The value being multiplied, 7 in our example, is called the 'base'. The Tiger Algebra multiplies exponents, showing you the step by step solution. <br> For example, you can enter	Les exposants sont une façon plus courte de noter la multiplication répétée d'un même terme par lui-même. <br> Par exemple <math>777*7=7^4</math> <br> L’exposant, 4 dans l’exemple ci-dessus, nous indique combien de fois la valeur est multipliée. La valeur multipliée, 7 dans notre exemple, est appelée la « base ». Tiger Algebra multiplie les exposants et te montre la solution étape par étape. <br> Par exemple, tu peux entrer

Key	English	French
IntegerToRomanNumeralStep1Title1	Identify integer	Normaliser et lire l entier
IntegerToRomanNumeralStep2TextUnit1	Split the number into place values: <math>{placeValues}</math>, then map each part to Roman symbols: <math>{romanPieces}</math>.	<math>{placeValues}</math>, <math>{romanPieces}</math>.
IntegerToRomanNumeralStep2Title1	Decompose place values	Decomposer en valeurs de position
IntegerToRomanNumeralStep3TextUnit1	Combine the mapped symbols to get the canonical Roman numeral: <math>{integerValue}</math> → <math>{romanResult}</math>.	<math>{integerValue}->{romanResult}</math>.
IntegerToRomanNumeralStep3Title1	Build Roman numeral	Assembler les symboles romains
JoinBenefit1	Live Help	Aide en direct
JoinBenefit2	Earn rewards and prizes	Gagne des récompenses et des prix
JoinBenefit3	Customize everything	Personnalise tout
JoinCommunityHeadline	Join a community with 26,970,008 solved equations	Rejoignez une communauté avec 26,970,008 équations résolues
KeyboardInputMode	Keyboard	Clavier
LatestRelatedDrillsSolved	Latest Related Drills Solved	Derniers exercices connexes résolus
LatestSolved	Latest Solved	Dernier exercice résolu
LatestTopics	Topics	Sujets
LatestVideoSectionTitle	Latest Videos	Dernières Vidéos
LCMCalculator	Least common multiple (LCM)	Calculatrice de PPCM
LCMCalculatorContent	The least common multiple (LCM), sometimes called the lowest common multiple or least common divisor (LCD), is the smallest number that two or more selected numbers can divide evenly into without a remainder. For example, <math>21</math> is the LCM of <math>3</math> and <math>7</math> because it is the lowest number that they both divide evenly into.<br><br> There are several ways to find the LCM: listing the multiples of each number, prime factorization, the cake or ladder method, the division method, or using the greatest common factor.<br><br> The Tiger Algebra LCM calculator not only finds the LCM but also shows the individual steps involved in doing so, helping you better understand and remember the process!	Le plus petit commun multiple (PPCM), parfois appelé le plus petit commun multiple ou le plus petit commun diviseur, est le plus petit nombre pouvant être divisé par deux ou plusieurs nombres sélectionnés sans reste. Par exemple, <math>21</math> est le PPCM de <math>3</math> et <math>7</math> car c'est le plus petit nombre pouvant être divisé par eux sans reste.<br><br> Il y a plusieurs façons de trouver le PPCM : en énumérant les multiples de chaque nombre, en factorisant les nombres premiers, en utilisant la méthode du cake ou de l'échelle, la méthode de la division ou en utilisant le plus grand facteur commun.<br><br> La calculatrice Tiger Algebra PPCM trouve non seulement le PPCM, mais montre les différentes étapes impliquées dans ce processus, t’aidant ainsi à mieux comprendre et te souvenir du processus !
LCMCalculatorFormattingGuide	<p>The least common multiple is the smallest number that is divisible by all other numbers in a sequence.</p> <p>For example, if you have a sequence 1, 2, 3, the Least Common Multiple is 6.</p> <p>lcm(1, 2, 3) = 6</p> <br> <h4>Common issues:</h4> <p>Don’t forget to write “lcm,” “least common multiple” or “LCM.”</p> <br> <h3>Examples</h3> <ul> <li><a href=/en/solution/least-common-multiple/lcm%20(24,20,36)/">lcm(24, 20, 36)</a></li> <li><a href="/en/solution/least-common-multiple/LCM%20of%2040%2048%2035/">LCM of 40 48 35</a></li> <li><a href="/en/solution/least-common-multiple/14,22,66LCM/">14,22,66LCM</a></li> <li><a href="/en/solution/least-common-multiple/least%20common%20multiple%201,2,3/">least common multiple 1,2,3</a></li> <li><a href="/en/solution/least-common-multiple/Lcm(21,35,20)/">Lcm(21,35,20)</a></li> </ul>	<p>Le multiple commun le plus petit est le plus petit nombre divisible par tous les autres nombres d'une séquence.</p> <p>Par exemple, si vous avez une séquence de 1, 2, 3, le multiple commun le plus petit est 6.</p> <p>pgcd(1, 2, 3) = 6</p> <br> <h4>Problèmes communs :</h4> <p>N'oubliez pas d'écrire “pgcd”, “plus grand commun diviseur” ou “PGCD”.</p> <br> <h3>Exemples</h3> <ul> <li><a href=/en/solution/least-common-multiple/pgcd%20(24,20,36)/">pgcd(24, 20, 36)</a></li> <li><a href="/en/solution/least-common-multiple/PGCD%20de%2040%2048%2035/">PGCD de 40 48 35</a></li> <li><a href="/en/solution/least-common-multiple/14,22,66pgcd/">14,22,66pgcd</a></li> <li><a href="/en/solution/least-common-multiple/plus%20grand%20commun%20diviseur%201,2,3/">plus grand commun diviseur 1,2,3</a></li> <li><a href="/en/solution/least-common-multiple/pgcd(21,35,20)/">pgcd(21,35,20)</a></li> </ul>
LearnMoreWithTiger	Learn more with Tiger	Apprendre plus avec Tiger
LeastCommonMultipleSolver	Least common multiple (LCM) by prime factorization	Plus petit commun multiple (PPCM) par factorisation en nombres premiers
LeastCommonMultipleSolver.plugin	the least common multiple (LCM) by prime factorization	Plus petit commun multiple (PPCM) par factorisation en nombres premiers
LeastCommonMultipleSolverSolverSpokenDescription	You entered an least common multiple problem!	Tu as entré un problème de plus petit commun multiple !
LeastCommonMultipleSolverStep1SingleNumber	The LCM (Least Common Multiple) of a single number is the number itself. Therefore, the LCM is {StepFinalResult}. <br><br> To find the LCM (Least Common Multiple) of a number, we need at least two numbers to compare. Could you please provide another number?	Le PPCM (Plus Petit Commun Multiple) d'un nombre unique est le nombre lui-même. Par conséquent, le PPCM est {StepFinalResult}. <br><br> Pour trouver le PPCM (Plus Petit Commun Multiple) d'un nombre, nous avons besoin d'au moins deux nombres à comparer. Pourriez-vous fournir un autre nombre?
LeastCommonMultipleSolverStep2Spoken	We continue, by counting how often each prime factor appears.	Nous continuons en comptant la fréquence à laquelle chaque facteur premier apparaît.
LeastCommonMultipleSolverStep2TableColLCMmax	Max. occurrence	Occurrence max.
LeastCommonMultipleSolverStep2TableColPrimeFactor	Prime factor	Facteur premier
LeastCommonMultipleSolverStep2TableTextUnit1	The prime {Step2AllFactorsWithOneOccurrenceList.Count: factor \| factors } {Step2AllFactorsWithOneOccurrenceList:{ListElement}\|, \| and } {Step2AllFactorsWithOneOccurrenceList.Count: occurs \| occur } one time.	Le(s) {Step2AllFactorsWithOneOccurrenceList.Count: facteur \| facteurs } premier(s) {Step2AllFactorsWithOneOccurrenceList:{ListElement}\|, \| et } {Step2AllFactorsWithOneOccurrenceList.Count: apparait \| apparaissent } une fois.
LeastCommonMultipleSolverStep2TableTextUnit2	The prime {Step2AllFactorsWithMoreOccurrencesList.Count: factor \| factors } {Step2AllFactorsWithMoreOccurrencesList:{ListElement}\|, \| and } {Step2AllFactorsWithMoreOccurrencesList.Count: occurs \| occur } more than once.	Le(s) {Step2AllFactorsWithOneOccurrenceList.Count: facteur \| facteurs } premier(s) {Step2AllFactorsWithOneOccurrenceList:{ListElement}\|, \| et } {Step2AllFactorsWithOneOccurrenceList.Count: apparait \| apparaissent } plus d’une fois.
LeastCommonMultipleSolverStep2TableTextUnit3	The prime {Step2AllFactorsWithOneOccurrenceList.Count: factor \| factors } {Step2AllFactorsWithOneOccurrenceList:{ListElement}\|, \| and } {Step2AllFactorsWithOneOccurrenceList.Count: occurs \| occur } one time, while {Step2AllFactorsWithMoreOccurrencesList:{ListElement}\|, \| and } {Step2AllFactorsWithMoreOccurrencesList.Count: occurs \| occur } more than once.	Le(s) {Step2AllFactorsWithOneOccurrenceList.Count: facteur \| facteurs } premier(s) {Step2AllFactorsWithOneOccurrenceList:{ListElement}\|, \| et } {Step2AllFactorsWithOneOccurrenceList.Count: apparait \| apparaissent } une fois, tandis que {Step2AllFactorsWithMoreOccurrencesList:{ListElement}\|, \| et } {Step2AllFactorsWithMoreOccurrencesList.Count: apparait \| apparaissent } plus d’une fois.
LeastCommonMultipleSolverStep2TableTitle	Determine the maximum number of times each prime factor ({Step2AllUniquePrimeFactorsList:{ListElement}\|, }) occurs in the factorization of the given numbers:	Déterminer le nombre maximum de fois où chaque facteur premier ({Step2AllUniquePrimeFactorsList:{ListElement}\|, }) apparaît dans la factorisation des nombres donnés :
LeastCommonMultipleSolverStep2Title	Build a prime factors table	Construire un tableau des facteurs premiers

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Le plus petit commun multiple (PPCM), parfois appelé le plus petit commun multiple ou le plus petit commun diviseur, est le plus petit nombre pouvant être divisé par deux ou plusieurs nombres sélectionnés sans reste. Par exemple, <math>21</math> est le PPCM de <math>3</math> et <math>7</math> car c'est le plus petit nombre pouvant être divisé par eux sans reste. 

Il y a plusieurs façons de trouver le PPCM : en énumérant les multiples de chaque nombre, en factorisant les nombres premiers, en utilisant la méthode du cake ou de l'échelle, la méthode de la division ou en utilisant le plus grand facteur commun. 

La calculatrice Tiger Algebra PPCM trouve non seulement le PPCM, mais montre les différentes étapes impliquées dans ce processus, t’aidant ainsi à mieux comprendre et te souvenir du processus !

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LCMCalculatorContent

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10 hours ago

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core/TigerAlgebra.Localization/Resources/TigerSharedResource.fr.resx, string 478