Unit-8 | HCF and LCM | Class-10


Overview: HCF stands for Highest Common Factor and is the common factor in all the given expressions. LCM stands for Lowest Common Multiples and is the lowest multiple that is common in the multiplication table of the given expressions. Here, we will learn to find HCF and LCM for Algebraic Expressions.


Contents

HCF 

LCM



HCF and LCM


Highest Common Factor (H.C.F.):

Factors are the building blocks of any expression. When we multiply two or more factors, we get an algebraic expression. 

When we take two or more such expressions and find the factors of each, the common factors in all the expressions are said to be the Highest Common Factor of those expressions. 

For example:

Let us take three expressions:
i) (a+b)
ii) a²-b² = (a+b)(a-b)
iii) a³+b³ = (a+b)(a²-ab+b²)

As we can see in the above three expressions, (a+b) is a common factor of each expressions. So, our Highest Common Factor of the above expressions is (a+b).

Remember these things while finding the HCF of given expressions:
  • Factorize the given expressions as much as possible.
  • Analyze the common factors that are common in all given expressions.
  • Write only the common factors and you will get your answer.
While finding HCF, you might want to know these important factorization formulae:
  • (a² -b²) = (a+b) (a-b)
  • (a³ -b³) = (a-b) (a² +ab +b²)
  • (a³ +b³) = (a+b) (a² -ab +b²)
  • (a³ -b³) = (a-b)³ + 3ab(a-b)
  • (a³ +b³) = (a+b)³ - 3ab(a+b)
Note: HCF of given expressions can be found out using both factorisation and division method.

See some solved examples of HCF:

Link: (a²-2a) and (a⁴-8a)

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Lowest Common Multiples (L.C.M.):

Every terms in Algebra can be multiplied. The Lowest Common Multiple is the first or the lowest common multiple of the given terms.

Let us take an example:

We have,
x⁶. Multiples of x⁶ are x⁶, x⁷, x⁸,x⁹, ...
x⁴. Multiples of x⁴ are x⁴, x⁵,x⁶, x⁷, ...

As you can see in the above multiples of x⁶ and x⁴, the lowest or the first common multiple is x⁶. So, Lowest Common Multiple of x⁶ and x⁴ is x⁶.

Take another example:

We have,
4. Multiples of 4 are 4, 8, 12, 16, 20, ...
8. Multiples of 8 are 8, 16, 24, 32, 40, ...

The first and the lowest common multiple is 8.

Remember these things while finding the LCM of the given expressions:
  • Factorize the given expressions as much as possible.
  • Write down the common factors only. If you have more than two expressions, factors common in at least two expressions can be termed as common factors. 
  • Now, multiply the common factors by the rest factors. 
  • Write the product in formula form, if possible.
Let us take an example:

We have,
1. (a+b)
2. (a² -b²) = (a+b)(a-b)
3. (a³ +b³) = (a+b) (a²-ab +b²)

As we discussed, we already factorised the given expressions.

Now, let's write the common factors only.
LCM = (a+b)

Then, multiply the rest factors with the common factor.
LCM = (a+b)(a-b)(a²-ab+b²)

Again, write them in the formula form.
LCM = (a³+b³)(a-b) or (a²-b²)(a²-ab+b²)


Check these solutions. You can see the solutions by click on "Find the HCF" written before every questions. 

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