Tiger Algebra/core — Portuguese @ 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	Portuguese
FactoringPolynomialsHavingMoreThan3Terms	Factoring polynomials with four or more terms	Fatorar polinómios com quatro termos ou mais
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>	Uma maneira simples de abordar a fatoração de um polinómio com quatro termos ou mais consiste em agrupá-lo em conjuntos de dois. Este método implica a análise desses conjuntos para ver se uma técnica em particular pode ser aplicada a ambos. Uma técnica para começar consiste em verificar se é possível encontrar o maior fator comum (MFC) entre um conjunto de dois termos. Se não for possível encontrar o MFC, os polinómios podem ser agrupados de outra forma e analisados por outra técnica. Existe sempre a possibilidade de o polinómio ser primo e não poder ser fatorado.
FactorTrinomials	Factor trinomials	Fatorar trinómios
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>	O Tiger Algebra mostra-te, passo a passo, como fatorar trinómios de acordo com diversos métodos.
FindAreaofTrianglegivenbyits3sides	Find area of triangle given by its 3 sides	Encontrar a área de um triângulo com base nos seus 3 lados
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>	Para encontrar a área de um triângulo com base nos seus 3 lados, introduz os tamanhos dos três lados separados por vírgulas e entre parêntesis. O Tiger Algebra irá mostrar-te a solução passo a passo. Entrada de exemplo: <math>(3,4,5)</math>
FindPrimeFactors	Find prime factors	Encontrar os fatores primos
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>	Um número primo só pode ser dividido por um ou por si mesmo. Os números inteiros compostos (não primos) podem ser escritos como um produto dos seus fatores primos. Introduz os fatores (qualquer número) ou primos (qualquer número) na calculadora de álgebra e o Tiger irá encontrar os seus fatores primos.
GCFCalculator	Greatest common factor	Maior fator comum
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.	O maior fator comum (MFC), por vezes denominado máximo fator comum (MFC) ou maior divisor comum (MDC), é o maior número inteiro positivo pelo qual pode ser dividido um conjunto de números inteiros. Por exemplo, o maior número pelo qual é possível dividir 12, 24 e 32 é 4, pelo que o seu maior fator comum é 4. Da mesma forma, o maior número pelo qual é possível dividir 3, 5 e 10 é 1, pelo que o respetivo maior fator comum é 1. <br><br> Existem duas formas de encontrar o maior fator comum: criar uma lista de fatores de cada número e a fatoração prima. <br><br> <b>Método 1: criar uma lista de fatores de cada número</b><br> Cria uma lista de todos os fatores de cada número e identifica o fator maior (superior) que todos os números têm em comum. <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 é o fator maior que os números têm em comum, pelo que é o maior fator comum. <br><br> <b>Método 2: fatoração prima</b><br> Utiliza uma árvore de fatores para identificar os fatores primos (fatores que são números primos) de cada número. Identifica os fatores primos que todos os números têm em comum e multiplica-os para obteres o maior fator comum. <br> <img src="/images/gcfTree1.png" alt="Prime factorization tree"><br> Os fatores primos que todos os números têm em comum são 2 e 2. Multiplica-os para obteres o maior fator comum, o 4.
GeometricSeries	Geometric series	Séries geométricas
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>	Na matemática, uma série geométrica é uma série com um rácio constante entre termos sucessivos. Por exemplo, a série <math>1,2,4,8</math> é geométrica porque cada termo sucessivo pode ser obtido multiplicando o termo anterior por <math>2</math>.
IdentifyingPerfectCubes	Identifying perfect cubes	Identificar cubos perfeitos
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>	Um cubo perfeito é um cubo de outro polinómio. O solucionador Tiger Algebra encontra cubos perfeitos e mostra-te as soluções passo a passo.
LCMCalculator	Least common multiple (LCM)	Calculadora de MMC
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!	O mínimo múltiplo comum (MMC), por vezes denominado menor múltiplo comum ou mínimo divisor comum (MDC) é o número mais pequeno que pode ser dividido sem resto por dois ou mais números selecionados. Por exemplo, <math>21</math> é o MMC de <math>3</math> e <math>7</math>, pois é o número mais baixo que pode ser uniformemente dividido sem resto por eles.<br><br> Existem várias formas de encontrar o MMC: criar uma lista de múltiplos de cada número; fatoração prima, o método do bolo ou da escada ou utilizando o maior fator comum.<br><br> A calculadora de MMC do Tiger Algebra não só encontra o MMC, como também mostra os passos individuais necessários para o encontrar, ajudando-te a compreender melhor e a recordar o processo!
LinearEquationswithOneUnknown	Linear equations with one unknown	Equações lineares com uma incógnita
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>	A aplicação principal de equações lineares centra-se na resolução de problemas nos quais uma variável desconhecida, normalmente (mas nem sempre) x, depende de uma constante conhecida.<br>Resolvemos equações lineares ao isolar a variável desconhecida num lado da equação e ao simplificar o resto da equação. Ao simplificar, <b>o que for feito de um lado da equação, também tem de ser feito do outro lado</b>.<br><br>Uma equação de:<br> <math>ax+b=0</math><br> em que <math>a</math> e <math>b</math> são as constantes e <math>x</math> é a variável desconhecida, é uma equação linear típica com um elemento desconhecido. Para resolver o <math>x</math> neste exemplo, primeiro teríamos de o isolar ao subtrair o <math>b</math> de ambos os lados da equação. Em seguida, teríamos de dividir ambos os lados da equação por <math>a</math>, resultando numa resposta de:<br><math>x = -b/a</math>
LinearEquationswithThreeUnknowns	Linear equations with three unknowns	Equações lineares com três incógnitas
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.	Um sistema linear de três equações com três variáveis é qualquer sistema que pode ser escrito na forma.
LinearEquationswithTwoUnknowns	Linear equations with two unknowns	Equações lineares com duas incógnitas
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	Um sistema linear de duas equações com duas variáveis é qualquer sistema que pode ser escrito na forma. <br><br> <math>ax+by=p</math> <br> <math>cx+dy=q</math> <br><br> em que as constantes (a,b,c e d) podem ser zero desde que cada equação tenha, pelo menos, uma variável (x ou y). Além disso, para o sistema ser denominado de linear, as variáveis só podem estar na primeira potência, estar apenas no numerador e não existirem produtos de variáveis noutras equações. <br><br> Uma solução para um sistema de equações é um valor de x e um valor de y que, quando substituídos nas equações, satisfaçam simultaneamente ambas as equações. <br><br> Separa as tuas equações com um ponto e vírgula ";" quando as introduzes no Tiger
LinearEquationsWithFourUnknowns	Linear equations with four unknowns	Equações lineares com quatro incógnitas
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>	Um grupo de quatro equações lineares com quatro fatores desconhecidos forma um sistema de equações. Resolver este sistema significa encontrar o valor dos fatores desconhecidos de modo a verificar todas as equações no sistema. O conceito geral por trás da resolução de um sistema de equações combina equações de tal forma que o número de variáveis seja reduzido. Tal pode ser feito por substituição ou eliminação (também designado redução de linha), e também por representação gráfica ou utilizando matrizes.
OtherFactorizations	Other factorizations	Outras fatorações
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>A fatorização envolve a decomposição de um número ou expressão algébrica em seus fatores constituintes. Além das técnicas comuns de fatorização como a fatorização em primos e a fatorização polinomial, existem outros métodos especializados utilizados em contextos específicos.</p> <h2>Técnicas Especializadas</h2> <ul> <li><strong>Diferença de quadrados:</strong> Fatorar expressões da forma \(a^2 - b^2\).</li> <li><strong>Soma ou diferença de cubos:</strong> Fatorar expressões da forma \(a^3 + b^3\) ou \(a^3 - b^3\).</li> <li><strong>Trinômios quadráticos:</strong> Fatorar expressões quadráticas da forma \(ax^2 + bx + c\).</li> </ul> <h2>Aplicações</h2> <p>Essas técnicas especializadas de fatorização são comumente usadas em manipulações algébricas, simplificações e solução de equações.</p> </div>
SolvingLinearEquationsBySubstitution	Solving linear equations by substitution	Resolver equações lineares por substituição
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>	A substituição é um dos métodos para resolver equações lineares. É uma boa opção se houver uma variável com um coeficiente de 1. Este método implica resolver uma das equações para uma variável e, em seguida, substituir essa expressão noutra equação para resolver outra variável. Tal irá produzir uma equação única com uma variável, que pode, depois, ser resolvida algebricamente.
LinearInequalities	Linear inequalities	Desigualdades lineares
MultiplyingExponents	Multiplying exponents	Multiplicar expoentes
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	Os expoentes são uma forma mais curta de indicar a multiplicação repetida da mesma coisa por ela mesma. <br> Por exemplo <math>7 x 7 x 7 x 7=7^4</math> <br> O "expoente", 4 no exemplo acima, indica-nos quantas vezes o valor é multiplicado. O valor multiplicado, 7 no nosso exemplo, é chamado de "base". O Tiger Algebra multiplica o expoente, mostrando-te a solução passo a passo. <br> Por exemplo, podes introduzir

Key	English	Portuguese
IntegerToRomanNumeralStep1Title1	Identify integer	Normalize e leia o numero inteiro
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	Decomponha por valores posicionais
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	Combine os simbolos romanos
JoinBenefit1	Live Help	Ajuda em tempo real
JoinBenefit2	Earn rewards and prizes	Ganhar recompensas e prémios
JoinBenefit3	Customize everything	Personalizar tudo
JoinCommunityHeadline	Join a community with 26,970,008 solved equations	Junte-se a uma comunidade com 26.970.008 equações resolvidas
KeyboardInputMode	Keyboard	Teclado
LatestRelatedDrillsSolved	Latest Related Drills Solved	Últimos exercícios relacionados resolvidos
LatestSolved	Latest Solved	Últimos resolvidos
LatestTopics	Topics	Tópicos
LatestVideoSectionTitle	Latest Videos	Últimos Vídeos
LCMCalculator	Least common multiple (LCM)	Calculadora de MMC
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!	O mínimo múltiplo comum (MMC), por vezes denominado menor múltiplo comum ou mínimo divisor comum (MDC) é o número mais pequeno que pode ser dividido sem resto por dois ou mais números selecionados. Por exemplo, <math>21</math> é o MMC de <math>3</math> e <math>7</math>, pois é o número mais baixo que pode ser uniformemente dividido sem resto por eles.<br><br> Existem várias formas de encontrar o MMC: criar uma lista de múltiplos de cada número; fatoração prima, o método do bolo ou da escada ou utilizando o maior fator comum.<br><br> A calculadora de MMC do Tiger Algebra não só encontra o MMC, como também mostra os passos individuais necessários para o encontrar, ajudando-te a compreender melhor e a recordar o 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>O mínimo múltiplo comum é o menor número que é divisível por todos os outros números em uma sequência.</p> <p>Por exemplo, se você tem uma sequência 1, 2, 3, o mínimo múltiplo comum é 6.</p> <p>mmc(1, 2, 3) = 6</p> <br> <h4>Problemas comuns:</h4> <p>Não esqueça de escrever “mmc,” “mínimo múltiplo comum” ou “MMC.”</p> <br> <h3>Exemplos</h3> <ul> <li><a href=/en/solution/least-common-multiple/mmc%20(24,20,36)/">mmc(24, 20, 36)</a></li> <li><a href="/en/solution/least-common-multiple/MMC%20de%2040%2048%2035/">MMC de 40 48 35</a></li> <li><a href="/en/solution/least-common-multiple/14,22,66MMC/">14,22,66MMC</a></li> <li><a href="/en/solution/least-common-multiple/mmc%20comum%201,2,3/">mínimo múltiplo comum 1,2,3</a></li> <li><a href="/en/solution/least-common-multiple/Mmc(21,35,20)/">Mmc(21,35,20)</a></li> </ul>
LearnMoreWithTiger	Learn more with Tiger	Aprende mais com o Tiger
LeastCommonMultipleSolver	Least common multiple (LCM) by prime factorization	Mínimo múltiplo comum (MMC) por fatoração prima
LeastCommonMultipleSolver.plugin	the least common multiple (LCM) by prime factorization	o mínimo múltiplo comum (MMC) por fatoração prima
LeastCommonMultipleSolverSolverSpokenDescription	You entered an least common multiple problem!	Introduziste um problema de mínimo múltiplo comum!
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?	O MMC (Mínimo Múltiplo Comum) de um único número é o próprio número. Portanto, o MMC é {StepFinalResult}. <br><br> Para encontrar o MMC (Mínimo Múltiplo Comum) de um número, precisamos de pelo menos dois números para comparar. Você poderia fornecer outro número?
LeastCommonMultipleSolverStep2Spoken	We continue, by counting how often each prime factor appears.	Continuamos contando a frequência com que cada fator primo aparece.
LeastCommonMultipleSolverStep2TableColLCMmax	Max. occurrence	Ocorrência máx.
LeastCommonMultipleSolverStep2TableColPrimeFactor	Prime factor	Fator primo
LeastCommonMultipleSolverStep2TableTextUnit1	The prime {Step2AllFactorsWithOneOccurrenceList.Count: factor \| factors } {Step2AllFactorsWithOneOccurrenceList:{ListElement}\|, \| and } {Step2AllFactorsWithOneOccurrenceList.Count: occurs \| occur } one time.	{Step2AllFactorsWithOneOccurrenceList.Count: Fator \| Fatores } primo(s) {Step2AllFactorsWithOneOccurrenceList:{ListElement}\|, \| e } {Step2AllFactorsWithOneOccurrenceList.Count: ocorre \| ocorrem } uma vez.
LeastCommonMultipleSolverStep2TableTextUnit2	The prime {Step2AllFactorsWithMoreOccurrencesList.Count: factor \| factors } {Step2AllFactorsWithMoreOccurrencesList:{ListElement}\|, \| and } {Step2AllFactorsWithMoreOccurrencesList.Count: occurs \| occur } more than once.	{Step2AllFactorsWithMoreOccurrencesList.Count: Fator \| Fatores } primo(s) {Step2AllFactorsWithMoreOccurrencesList:{ListElement}\|, \| e } {Step2AllFactorsWithMoreOccurrencesList.Count: ocorre \| ocorrem } mais do que uma vez.
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: Fator \| Fatores } primo(s) {Step2AllFactorsWithOneOccurrenceList:{ListElement}\|, \| e } {Step2AllFactorsWithOneOccurrenceList.Count: ocorre \| ocorrem } uma vez, enquanto {Step2AllFactorsWithMoreOccurrencesList:{ListElement}\|, \| e } {Step2AllFactorsWithMoreOccurrencesList.Count: ocorre \| ocorrem } mais do que uma vez.
LeastCommonMultipleSolverStep2TableTitle	Determine the maximum number of times each prime factor ({Step2AllUniquePrimeFactorsList:{ListElement}\|, }) occurs in the factorization of the given numbers:	Determinar o número máximo de vezes que cada fator primo ({Step2AllUniquePrimeFactorsList:{ListElement}\|, }) ocorre na fatoração dos números indicados:
LeastCommonMultipleSolverStep2Title	Build a prime factors table	Elaborar uma tabela de fatores primos

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O mínimo múltiplo comum (MMC), por vezes denominado menor múltiplo comum ou mínimo divisor comum (MDC) é o número mais pequeno que pode ser dividido sem resto por dois ou mais números selecionados. Por exemplo, <math>21</math> é o MMC de <math>3</math> e <math>7</math>, pois é o número mais baixo que pode ser uniformemente dividido sem resto por eles. 

Existem várias formas de encontrar o MMC: criar uma lista de múltiplos de cada número; fatoração prima, o método do bolo ou da escada ou utilizando o maior fator comum. 

A calculadora de MMC do Tiger Algebra não só encontra o MMC, como também mostra os passos individuais necessários para o encontrar, ajudando-te a compreender melhor e a recordar o processo!

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LCMCalculatorContent

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9 hours ago

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