H.C.F. and L.C.M. of Algebraic Expressions: Definitions, Methods, Examples

H.C.F. and L.C.M. of Algebraic Expressions: H.C.F and L.C.M are the two essential terms for the highest common factor and least common multiple, respectively. The H.C.F is the largest factor of the two or more numbers, dividing the numbers with no remainder. On the contrary, the L.C.M of two or more numbers is the smallest number divisible by the given numbers. 

We can find H.C.F and L.C.M of the algebraic expressions. We could find H.C.F using the factoring method and long division method, and L.C.M by factoring method for numbers and algebraic expressions. Let us learn the methods of H.C.F and L.C.M of algebraic expressions.

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Highest Common Factor (H.C.F) of Algebraic Expressions

The highest common factor (H.C.F) of a set of algebraic expressions is the largest degree of an algebraic expression that divides all the given algebraic expressions completely. H.C.F is also known as GCF (Greatest Common Factor) or GCD (Greatest Common Divisor).

Finding H.C.F Using Prime Factorisation Method

The prime factorisation method is also known as the factor tree method. Now, follow the given points to find out the H.C.F:

1. Write each of the algebraic expressions as a product of its prime factors.
2. Then you have to make a list of the common factors for the algebraic expressions.
3. The product of the prime factors is considered as H.C.F.

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Example 1
\(2x(x+1,) 4x\)
Here, two algebraic expressions are \(2x(x+1)\) and \(4x.\)
Step 1: \(2x(x+1)=2×x×(x+1).\) The factors of \(2x(x+1)=2, x, (x+1)\)
Step 2: \(4x=2×2×x.\) Thus, the factors of \(4x=2, 2, x.\)
Step 3: The common factors of both expressions are \(2, x\)
Step 4: Hence, the H.C.F is \(2x\)

Example 2
\(10\,pq,\,pqr,\,5\,{p^2}\)
Here, three algebraic expressions are \(10\,pq,\,pqr,\,5\,{p^2}.\)
Step 1: \(10\,pq=2×5×p×q.\) The factors of \(10\,pq=2, 5, p, q\)
Step 2: \(pqr=p×q×r.\) Thus, the factors of \(pqr=p, q, r\)
Step 3: \(5\,{p^2} = 5 \times p \times p.\) Thus, the factors of \(5\,{p^2} = 5,p,p\)
Step 3: The common factor of three algebraic expressions are \(p.\)
Step 4: Hence, the H.C.F is \(p.\)

Finding H.C.F Using the Long Division Method

This method divides the algebraic expression with a greater degree by the algebraic expression with a smaller degree and checks the remainder. Then, we make the remainder of the above step as the divisor and the divisor of the above step as the dividend and do the long division again. 

We keep on doing the long division until we get the remainder as \(0,\) and the last divisor will be the H.C.F of those two algebraic expressions. We can find H.C.F of more than two algebraic expressions using the same method. Let’s understand this method using an example.

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Example 3: Find the H.C.F of \({a^3} + 4{a^2} + 4a + 1,{a^4} + 3{a^3} + 2{a^2} + 3a + 1\)
Solution:

Step 1: Organise the terms of the dividend and divisor in descending order of their degrees. \({a^4} + 3{a^3} + 2{a^2} + 3a + 1\) is the algebraic expression with a degree \(4\) and \({a^3} + 4{a^2} + 4a + 1\) is the algebraic expression with degree 3. So, we will consider  \({a^3} + 4{a^2} + 4a + 1\) as divisor and \({a^4} + 3{a^3} + 2{a^2} + 3a + 1\) as a dividend.

Step 2: After finishing the first division, we get  \(2{a^2} + 6a + 2 = 2\left( {{a^2} + 3a + 1} \right)\) as remainder. To simplify the next step, we will consider the remainder as \({a^2} + 3a + 1\) instead of \(2{a^2} + 6a + 2.\)

Step 3: Again, we need to perform another division. We will make remainder that is \({a^2} + 3a + 1\)  of step 2 as the divisor and the divisor of step 2 that is  \({a^3} + 4{a^2} + 4a + 1\) as a dividend. We will repeat the method until we get \(0\) in the remainder. Here, we get \(0\) as the remainder.

Step 4: The last divisor is \({a^2} + 3a + 1,\) where we get \(0\) as remainder. Hence the H.C.F is \({a^2} + 3a + 1.\)

Lowest Common Factor (L.C.M) of Algebraic Expressions

L.C.M stands for lowest or least common multiple. In other words, the L.C.M of two or more algebraic expressions is the algebraic expression with the smallest degree divisible by all the given algebraic expressions.

Finding L.C.M Using Factoring Method

1. Write each of the algebraic expressions as a product of its prime factors.
2. Then you have to make a list of each factor with the highest power for the algebraic expressions.
3. Find the product of each factor with the highest power.
4. The product so obtained is the L.C.M of the given algebraic expressions.

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Example 4

\(6{x^2}y,2xy\)
Here, Two algebraic expressions are \(6{x^2}y,2xy.\)
Step 1: \(6{x^2}y = 2 \times 3 \times x \times x \times y.\) The factors of \(6{x^2}y = 2,3,x,x,y.\)
Step 2: \(2xy=2×x×y.\) Thus, the factors of \(2xy=2, x, y.\)
Step 3: The common multiple of both expressions are \(2, 3, x, x, y.\)
Step 4: Hence, the L.C.M is \(2 \times 3 \times x \times x \times y = 6{x^2}y.\)

Example 5

\(4{a^2}b,6ab,8a{b^2}\)
Here, three algebraic expressions are \(4{a^2}b,6ab,8a{b^2}\)
Step 1: \(4{a^2}b = 2 \times 2 \times a \times a \times b = {2^2} \times {a^2} \times b.\) The factors of \(4{a^2}b = 2 \times 2 \times a \times a \times b.\)
Step 2: \(6ab=2×3×a×b.\) Thus, the factors of \(6ab=2, 3,a, b.\)
Step 3: \(8a{b^2} = 2 \times 2 \times 2 \times a \times b \times b = {2^3} \times a \times {b^2}\)
Step 4: We will find the product of each factor with the highest power. Hence, the lowest common multiple of the expressions are \({2^3} \times 3 \times {a^2} \times {b^2} = 24{a^2}{b^2}.\)

Solved Examples – H.C.F. and L.C.M. of Algebraic Expressions

Q.1. Find the H.C.F of the following expressions using the factoring method
\(3a(2a + 1),6{(2a + 1)^2}.\)
Ans: Here, two algebraic expressions are
\(3a(2a + 1),6{(2a + 1)^2}\)
Step 1: \(6{(2a + 1)^2} = 2 \times 3 \times (2a + 1) \times (2a + 1).\). The factors of \(3a(2a+1)=3, a, (2a+1)\)
Step 2: \(6{(2a + 1)^2} = 2 \times 3 \times (2a + 1) \times (2a + 1).\) Thus, the factors of \(6{(2a + 1)^2} = 2,3,(2a + 1),(2a + 1).\)
Step 3: The common factors of both expressions are \(3, (2a+1)\)
Step 4: Hence, the H.C.F is \(3(2a+1)\)

Q.2. Find the H.C.F of the following expressions using the factoring method
\(5ab, 20abc,15bc.\)
Ans: Here, three algebraic expressions are \(5ab, 20abc,15bc\)
Step 1: \(5ab=5×a×b.\) The factors of \(5ab=5, a, b\)
Step 2: \(20abc=2×2×5×a×b×c.\) Thus, the factors of \(20abc=2, 2, 5, a, b, c.\)
Step 3: \(15bc=3×5×b×c.\) Thus, the factors of \(15bc=3, 5, b, c\)
Step 3: The common factors of three algebraic expressions are \(5, b.\) The product of the common factors is \(5b.\)
Step 4: Hence, the H.C.F is \(5b.\)

Q.3. Find the L.C.M of the following expressions using the factoring method
\(27y, 3xy.\)
Ans: Here, two algebraic expressions are \(27y, 3xy\)
Step 1: \(27y = 3 \times 3 \times 3 \times y = {3^3} \times y\) The factors of \(27y=3, 3,3, y.\)
Step 2: \(3xy=3×x×y.\) Thus, the factors of \(3xy=3, x, y.\)
Step 3: We will find the product of each factor with the highest power. The common multiple of both expressions is \(27, x, y.\)
Step 4: Hence, the L.C.M is \(27xy.\)

Q.4. Find the L.C.M of the following expressions using the factoring method
\({a^2}b,5ab,a{b^2}\)
Ans: Here, three algebraic expressions are\({a^2}b,5ab,a{b^2}\)
Step 1: \({a^2}b = a \times a \times b = {a^2} \times b.\) The factors of \({a^2}b = a,a,b\)
Step 2: \(5ab=5×a×b.\) Thus, the factors of \(5ab=5,a,\) b
Step 3: \(a{b^2} = a \times b \times b = a \times {b^2}.\) Thus, the factors of \(a{b^2} = a,b,b\)
Step 4: We will find the product of each factor with the highest power. Hence, the lowest common multiple of the expressions are \(5 \times {a^2} \times {b^2} = 5{a^2}{b^2}.\)

Q.5. Find the H.C.F of \({x^3} – 3{x^2} + 4x – 12,{x^4} + {x^3} + 4{x^2} + 4x\) using long division method
Ans:

Step 1: \({x^4} + {x^3} + 4{x^2} + 4x\) is the algebraic expression with degree \(4\) and \({x^3} – 3{x^2} + 4x – 12\) is the algebraic expression with degree 3. So, we will consider  \({x^3} – 3{x^2} + 4x – 12\) as divisor and \({x^4} + {x^3} + 4{x^2} + 4x\) as a dividend. To avoid the calculation complexity, we take common x from  \({x^4} + {x^3} + 4{x^2} + 4x\) that gives  \({x^3} + {x^2} + 4x + 4\) and we make it the dividend.
Step 2: After finishing the division, we get  \(\left( { – 4{x^2} – 16} \right)\) as remainder. To simplify the next step, we will consider the remainder as \(\left( {{x^2} + 4} \right)\) instead of  \(\left( { – 4{x^2} – 16} \right)\) as we took common as \(-4\) from it.
Step 3: Again, we need to perform another division. We will make remainder that is  \(\left( {{x^2} + 4} \right)\) of step 2 as the divisor and the divisor of step 2 that is \({x^3} – 3{x^2} + 4x – 12\) as a dividend. We will repeat the method until we get \(0\) in the remainder. Here, we get \(0\) as the remainder.
Step 4: The last divisor is \({x^2} + 4,\) where we get \(0\) as remainder. Hence the H.C.F is  \({x^2} + 4.\)

Summary

This article tells about the H.C.F and L.C.M of algebraic expressions and the methods of finding H.C.F and L.C.M of algebraic expressions. We learnt two methods of finding H.C.F of algebraic expression: the factoring method and the long division method and finding L.C.M using the factoring method. We solved some numerical problems related to the topic.

Relationship Between HCF and LCM

Frequently Asked Question (FAQ) – H.C.F. and L.C.M. of Algebraic Expressions

Q.1. How do you find the H.C.F and L.C.M of an algebraic expression?
Ans: The highest common factor (H.C.F) of a set of algebraic expressions is the largest degree of an algebraic expression that divides all the given algebraic expressions completely. H.C.F is also known as GCF (Greatest Common Factor) or GCD (Greatest Common Divisor).
L.C.M stands for lowest or least common multiple. In other words, the L.C.M of two or more algebraic expressions is the algebraic expression with the smallest degree divisible by all the given algebraic expressions.

Q.2. How do you find the L.C.M of a number with variables?
Ans: The L.C.M of a number and a variable is the product of them.
For example, L.C.M of \(x\) and \(5\) is \(5x.\)

Q.3. What is the long division method of two algebraic expressions?
Ans: This method divides the algebraic expression with a greater degree by the algebraic expression with a smaller degree and checks the remainder. Then, we make the remainder of the above step as the divisor and the divisor of the above step as the dividend and do the long division again.
We keep on doing the long division until we get the remainder as \(0,\) and the last divisor will be the H.C.F of those two algebraic expressions. We can find H.C.F of more than two algebraic expressions using the same method.

Q.4. What is H.C.F and L.C.M of numbers?
Ans: L.C.M stands for lowest or least common multiple. In other words, the L.C.M of two or more numbers is the smallest positive integer divisible by all the given numbers.
The greatest common factor (GCF or GCD or H.C.F) of a set of whole numbers is the largest positive integer that divides all the given numbers evenly with zero remainders. H.C.F stands for the highest common factor. Thus, H.C.F is also known as GCF (Greatest Common Factor) or GCD (Greatest Common Divisor).

Q.5. What is the product of H.C.F and L.C.M of two numbers?
Ans: The product of H.C.F and L.C.M is equal to the product of the two numbers.

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