Tiger Algebra/core — Italian @ 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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FactoringPolynomialsHavingMoreThan3Terms	Factoring polynomials with four or more terms	Scomposizione di polinomi con quattro o più termini
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>	Un modo semplice per affrontare la scomposizione di un polinomio con quattro o più termini è quello di raggrupparlo in insiemi di due fattori. Questo metodo implica l'analisi congiunta di questi insiemi per stabilire se è possibile adottare una particolare tecnica per entrambi. Una tecnica per verificare se è possibile calcolare il massimo comune divisore (MCD) tra un insieme di due termini. Se non è possibile calcolare il MCD, i polinomi potrebbero essere raggruppati in un altro modo ed esaminati al fine di stabilire un'altra tecnica. C'è sempre la possibilità che il polinomio sia un numero primo e che non possa essere scomposto.
FactorTrinomials	Factor trinomials	Scomposizione dei trinomi
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 ti mostra in modo dettagliato come scomporre i trinomi secondo diversi metodi.
FindAreaofTrianglegivenbyits3sides	Find area of triangle given by its 3 sides	Trova l'area del triangolo dato dai suoi 3 lati
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>	Per trovare l'area del triangolo dati i suoi 3 lati, inserisci le dimensioni dei tre lati separati da virgole e racchiusi tra parentesi. Tiger Algebra ti mostrerà la soluzione nei singoli passaggi. Esempio di immissione: <math>(3,4,5)</math>
FindPrimeFactors	Find prime factors	Individuazione dei fattori primi
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 numero primo è divisibile solo per uno o per se stesso. I numeri interi composti (non primi) possono essere scritti come prodotto dei rispettivi fattori primi. Digita i fattori (qualsiasi numero) o i numeri primi (qualsiasi numero) nella calcolatrice algebrica e Tiger troverà per te i rispettivi fattori primi.
GCFCalculator	Greatest common factor	Massimo comune divisore
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.	Il massimo comune divisore (MCD), talvolta chiamato anche massimo comune fattore (HCF) omassimo fattore comune (GCF), è il più grande intero positivo per il quale un insieme di numeri interi è divisibile. Per esempio, il più grande numero per il quale possono essere divisi 12, 24 e 32 è 4, quindi il loro massimo comune divisore è 4. Allo stesso modo, il più grande numero per il quale possono essere divisi 3, 5 e 10 è 1, quindi il loro massimo comune divisore è 1. <br><br> Ci sono due metodi per calcolare il massimo comune divisore: quello dell'elenco dei divisori di ogni numero e quello della scomposizione in fattori primi. <br><br> <b>Metodo 1: elencare i divisori di ciascun numero</b><br> Fai un elenco di tutti i fattori di ciascun numero e identifica il più grande tra i divisori che tutti i numeri hanno in comune. <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 è il più grande divisore comune ai numeri, pertanto è il massimo comune divisore. <br><br> <b>Metodo 2: scomposizione in fattori primi</b><br> Usa un albero dei fattori per identificare i fattori primi (fattori che sono numeri primi) di ciascun numero. Identifica i fattori primi che tutti i numeri hanno in comune e moltiplicali fra loro per ottenere il massimo comune divisore. <br> <img src="/images/gcfTree1.png" alt="Prime factorization tree"><br> I fattori primi che tutti i numeri hanno in comune sono 2 e 2. Moltiplicali fra loro per ottenere il massimo comune divisore, 4.
GeometricSeries	Geometric series	Serie geometriche
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>	In matematica, una serie geometrica è una serie tale per cui il rapporto tra due termini successivi è costante. Per esempio, la serie <math>1,2,4,8</math> è geometrica, perché ogni termine successivo può essere ottenuto moltiplicando il termine precedente per <math>2</math>.
IdentifyingPerfectCubes	Identifying perfect cubes	Identificazione di cubi perfetti
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 cubo perfetto è il cubo di un altro polinomio. Il risolutore Tiger Algebra trova i cubi perfetti e ti mostra le soluzioni nei singoli passaggi.
LCMCalculator	Least common multiple (LCM)	Calcolatrice mcm
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!	Il minimo comune multiplo (mcm), talvolta chiamato anche minimo comune denominatore (LCD) è il più piccolo numero equamente divisibile per due o più numeri selezionati senza dare un resto. Per esempio, <math>21</math> è il minimo comune multiplo di <math>3</math> e di <math>7</math> dal momento che è il più piccolo numero equamente divisibile per entrambi senza dare un resto. <br><br> Ci sono vari metodi per calcolare il mcm: fare un elenco dei multipli di ciascun numero, scomposizione in fattori primi, il metodo della torta o della scala, il metodo della divisione o l'impiego del massimo comune divisore.<br><br> La calcolatrice di mcm di Tiger Algebra non solo trova il mcm ma mostra anche i singoli passaggi dell'operazione, aiutandoti a capire meglio e a memorizzare il processo!
LinearEquationswithOneUnknown	Linear equations with one unknown	Equazioni lineari a una incognita
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>	L'applicazione principale delle equazioni lineari è la risoluzione di problemi in cui una variabile sconosciuta, solitamente (ma non sempre) x, dipende da una costante nota.<br>Risolviamo le equazioni lineari isolando la variabile sconosciuta su un lato dell'equazione e semplificando il resto dell'equazione. Quando si semplifica, <b>qualunque operazione si esegua su un lato dell'equazione, deve essere eseguita anche sull'altro lato</b>.<br><br>Un'equazione del tipo:<br> <math>ax+b=0</math><br> in cui <math>a</math> e <math>b</math> sono le costanti e <math>x</math> è la variabile sconosciuta, è una tipica equazione lineare ad una sola incognita. Per risolvere <math>x</math> in questo esempio, dovremmo prima isolare la variabile sottraendo <math>b</math> da entrambi i lati dell'equazione. Poi dovremmo dividere entrambi i lati dell'equazione per <math>a</math>, ottenendo come risultato:<br><math>x = -b/a</math>
LinearEquationswithThreeUnknowns	Linear equations with three unknowns	Equazioni lineari a tre incognite
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 sistema lineare di tre equazioni a tre variabili è un qualsiasi sistema che può essere scritto nella seguente forma:
LinearEquationswithTwoUnknowns	Linear equations with two unknowns	Equazioni lineari a due incognite
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 sistema lineare di due equazioni con due variabili è qualsiasi sistema che può essere scritto nella seguente forma: <br><br> <math>ax+by=p</math> <br> <math>cx+dy=q</math> <br><br> dove le costanti (a,b,c e d) possono essere zero, purché ogni equazione contenga almeno una variabile (x o y). Inoltre, affinché il sistema possa definirsi lineare, le variabili possono comparire solo alla prima potenza, solo al numeratore e nessuna delle equazioni deve contenere prodotti di variabili. <br><br> La soluzione di un sistema di equazioni è data da un valore di x e da un valore di y che, se sostituiti nelle equazioni, le soddisfano entrambe contemporaneamente. <br><br> Separa le tue equazioni con un punto e virgola ";" quando le inserisci in Tiger.
LinearEquationsWithFourUnknowns	Linear equations with four unknowns	Equazioni lineari a quattro incognite
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 gruppo di quattro equazioni lineari con quattro incognite forma un sistema di equazioni. Risolvere questo sistema significa calcolare il valore delle incognite che risolva tutte le equazioni del sistema. Il concetto generale alla base della risoluzione di un sistema di equazioni è quello di combinare le equazioni in modo tale da ridurre il numero di variabili. Ciò è possibile tramite sostituzione o eliminazione (chiamata anche riduzione delle righe), ma anche tramite grafici o usando matrici.
OtherFactorizations	Other factorizations	Altre scomposizioni in fattori primi
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>	La factorización implica descomponer un número o expresión algebraica en sus factores constituyentes. Además de las técnicas comunes de factorización como la factorización prima y la factorización polinómica, hay otros métodos especializados utilizados en contextos específicos. Métodos Especializados: Diferencia de cuadrados: Factoring expresiones de la forma \(a^2 - b^2\). Suma o diferencia de cubos: Factoring expresiones de la forma \(a^3 + b^3\) o \(a^3 - b^3\). Trinomios cuadráticos: Factoring expresiones cuadráticas de la forma \(ax^2 + bx + c\). Aplicaciones: Estas técnicas de factorización especializadas se utilizan comúnmente en manipulaciones algebraicas, simplificaciones y resolución de ecuaciones.
SolvingLinearEquationsBySubstitution	Solving linear equations by substitution	Risoluzione di equazioni lineari per sostituzione
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 sostituzione è uno dei metodi per risolvere le equazioni lineari. È un procedimento valido in presenza di una variabile con coefficiente pari a 1. Questo metodo consiste nel risolvere una delle equazioni per una variabile per poi sostituire quell'espressione in un'altra equazione per risolvere un'altra variabile. In questo modo si ottiene una singola equazione con una sola variabile, che può poi essere risolta algebricamente.
LinearInequalities	Linear inequalities	Disequazioni lineari
MultiplyingExponents	Multiplying exponents	Moltiplicazione degli esponenti
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	Gli esponenti rappresentano un modo più breve di indicare la moltiplicazione ripetuta dello stesso elemento per se stesso. <br> Per esempio <math>777*7=7^4</math> <br> L''esponente', 4 nell'esempio precedente, ci dice per quante volte viene moltiplicato il valore. Il valore che viene moltiplicato, 7 nel nostro esempio, è chiamato 'base'. Tiger Algebra ti mostrerà la soluzione per singoli passaggi. <br> Per esempio, puoi digitare

Key	English	Italian
IntegerToRomanNumeralStep1Title1	Identify integer	Normalizza e leggi l'intero
IntegerToRomanNumeralStep2TextUnit1	Split the number into place values: <math>{placeValues}</math>, then map each part to Roman symbols: <math>{romanPieces}</math>.	Scomponi il numero per ordini di grandezza: <math>{placeValues}</math>; simboli romani corrispondenti: <math>{romanPieces}</math>.
IntegerToRomanNumeralStep2Title1	Decompose place values	Scomponi per valori posizionali
IntegerToRomanNumeralStep3TextUnit1	Combine the mapped symbols to get the canonical Roman numeral: <math>{integerValue}</math> → <math>{romanResult}</math>.	Combina i simboli ottenuti: <math>{integerValue}->{romanResult}</math>.
IntegerToRomanNumeralStep3Title1	Build Roman numeral	Combina i simboli romani
JoinBenefit1	Live Help	Live Help
JoinBenefit2	Earn rewards and prizes	Guadagna ricompense e premi
JoinBenefit3	Customize everything	Personalizza tutto
JoinCommunityHeadline	Join a community with 26,970,008 solved equations	Join a community with 26,970,008 solved equations
KeyboardInputMode	Keyboard	Tastiera
LatestRelatedDrillsSolved	Latest Related Drills Solved	Ultimi esercizi correlati risolti
LatestSolved	Latest Solved	Ultimi risolti
LatestTopics	Topics	Argomenti
LatestVideoSectionTitle	Latest Videos	Ultimi Video
LCMCalculator	Least common multiple (LCM)	Calcolatrice mcm
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!	Il minimo comune multiplo (mcm), talvolta chiamato anche minimo comune denominatore (LCD) è il più piccolo numero equamente divisibile per due o più numeri selezionati senza dare un resto. Per esempio, <math>21</math> è il minimo comune multiplo di <math>3</math> e di <math>7</math> dal momento che è il più piccolo numero equamente divisibile per entrambi senza dare un resto. <br><br> Ci sono vari metodi per calcolare il mcm: fare un elenco dei multipli di ciascun numero, scomposizione in fattori primi, il metodo della torta o della scala, il metodo della divisione o l'impiego del massimo comune divisore.<br><br> La calcolatrice di mcm di Tiger Algebra non solo trova il mcm ma mostra anche i singoli passaggi dell'operazione, aiutandoti a capire meglio e a memorizzare il processo!
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>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>
LearnMoreWithTiger	Learn more with Tiger	Impara di più con Tiger
LeastCommonMultipleSolver	Least common multiple (LCM) by prime factorization	Minimo comune multiplo (mcm) mediante la scomposizione in fattori primi
LeastCommonMultipleSolver.plugin	the least common multiple (LCM) by prime factorization	il minimo comune multiplo (mcm) mediante la scomposizione in fattori primi
LeastCommonMultipleSolverSolverSpokenDescription	You entered an least common multiple problem!	Hai inserito un problema con minimo comune multiplo!
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?	Il MCM (Minimo Comune Multiplo) di un singolo numero è il numero stesso. Pertanto, il MCM è {StepFinalResult}. <br><br> Per trovare il MCM (Minimo Comune Multiplo) di un numero, abbiamo bisogno di almeno due numeri da confrontare. Potresti fornire un altro numero?
LeastCommonMultipleSolverStep2Spoken	We continue, by counting how often each prime factor appears.	Continuiamo, contando il numero di volte in cui è presente ogni fattore primo.
LeastCommonMultipleSolverStep2TableColLCMmax	Max. occurrence	Occorrenza massima
LeastCommonMultipleSolverStep2TableColPrimeFactor	Prime factor	Fattore primo
LeastCommonMultipleSolverStep2TableTextUnit1	The prime {Step2AllFactorsWithOneOccurrenceList.Count: factor \| factors } {Step2AllFactorsWithOneOccurrenceList:{ListElement}\|, \| and } {Step2AllFactorsWithOneOccurrenceList.Count: occurs \| occur } one time.	{Step2AllFactorsWithOneOccurrenceList.Count: Il fattore primo \| I fattori primi } {Step2AllFactorsWithOneOccurrenceList:{ListElement}\|, \| e } {Step2AllFactorsWithOneOccurrenceList.Count: è presente \| sono presenti } una volta.
LeastCommonMultipleSolverStep2TableTextUnit2	The prime {Step2AllFactorsWithMoreOccurrencesList.Count: factor \| factors } {Step2AllFactorsWithMoreOccurrencesList:{ListElement}\|, \| and } {Step2AllFactorsWithMoreOccurrencesList.Count: occurs \| occur } more than once.	{Step2AllFactorsWithMoreOccurrencesList.Count: Il fattore primo \| I fattori primi } {Step2AllFactorsWithMoreOccurrencesList:{ListElement}\|, \| e } {Step2AllFactorsWithMoreOccurrencesList.Count: è presente \| sono presenti } più di una volta.
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.	{Step2AllFactorsWithOneOccurrenceList.Count: Il fattore primo \| I fattori primi } {Step2AllFactorsWithOneOccurrenceList:{ListElement}\|, \| e } {Step2AllFactorsWithOneOccurrenceList.Count: è presente \| sono presenti } una sola volta, mentre {Step2AllFactorsWithMoreOccurrencesList:{ListElement}\|, \| e } {Step2AllFactorsWithMoreOccurrencesList.Count: è presente \| sono presenti } più di una volta.
LeastCommonMultipleSolverStep2TableTitle	Determine the maximum number of times each prime factor ({Step2AllUniquePrimeFactorsList:{ListElement}\|, }) occurs in the factorization of the given numbers:	Calcola il numero massimo di volte in cui ogni fattore primo ({Step2AllUniquePrimeFactorsList:{ListElement}\|, }) è presente nella scomposizione in fattori primi dei numeri dati:
LeastCommonMultipleSolverStep2Title	Build a prime factors table	Crea una tabella di fattori primi

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Il minimo comune multiplo (mcm), talvolta chiamato anche minimo comune denominatore (LCD) è il più piccolo numero equamente divisibile per due o più numeri selezionati senza dare un resto. Per esempio, <math>21</math> è il minimo comune multiplo di <math>3</math> e di <math>7</math> dal momento che è il più piccolo numero equamente divisibile per entrambi senza dare un resto.
 

Ci sono vari metodi per calcolare il mcm: fare un elenco dei multipli di ciascun numero, scomposizione in fattori primi, il metodo della torta o della scala, il metodo della divisione o l'impiego del massimo comune divisore. 

La calcolatrice di mcm di Tiger Algebra non solo trova il mcm ma mostra anche i singoli passaggi dell'operazione, aiutandoti a capire meglio e a memorizzare il processo!

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Glossary

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LCMCalculatorContent

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9 hours ago

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