Highest Common Factor and Lowest Common Factor Multiple
The Highest Common Factor (H.C.F) of two or more numbers is the largest number that divides evenly into both numbers. In other words, the H.C.F is the largest of all the common factors.The Lowest Common Multiple (L.C.M) is the smallest number that is a common multiple of two or more numbers.
Summary
The Highest Common Factor (H.C.F) of two or more numbers is the largest number that divides evenly into both numbers. In other words, the H.C.F is the largest of all the common factors.The Lowest Common Multiple (L.C.M) is the smallest number that is a common multiple of two or more numbers.
Things to Remember
- The H.C.F of two or more numbers is smaller than or equal to the smallest number of given numbers.
- The L.C.M of two or more numbers is greater than or equal to the greatest number of given numbers.
- The smallest number which is exactly divisible by x, y and z are L.C.M of x, y, z.
- If the H.C.F of the numbers a, b, c is K, then a, b, c can be written as multiples of K (Kx, Ky, Kz, where x, y, z are some numbers). K divides the numbers a, b, c, so the given numbers can be written as the multiples of K.
- If the H.C.F of the numbers a, b is K, then the numbers (a + b), (a -b) is also divisible by K. The numbers a and b can be written as the multiples of K.
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Highest Common Factor and Lowest Common Factor Multiple
Highest common factor (HCF)
Let's take two expressions xy and yz. Here, xy is the product of x and y and yz is the product of y and z. x and y are factors of xy and y and z are the factors of yz. y is factor of both the expressions. So, y is called the highest common factor (HCF) of the expressions xy and yz.
\( \boxed {Note: HCF\:\text {divides each of the given expression exactly} } \)
To find the HCF
- find the factors of the given expressions.
- choose the common factors of the expressions.
- express the HCF in the form of product.
| Note: If there is no any common factor in the given expression, HCF = 1 as 1 is the factor of any number. |
Lowest Common Multiple (LCM)
Let's take two expressions a2b - ab2and a3b - ab3.Factorizing the expressions,
Here,
\begin{align*} first \: expression &= a^2 b - ab^2 \\ &= ab (a -b) \end{align*}
\begin{align*} second \: expression &= a^3 b - ab^3 \\ &=ab (a^2 - b^2) \\ &= ab(a + b) \: (a - b)\\ \end{align*}
common factors = ab(a - b)
Remaining factors = (a + b)
\begin{align*} LCM &= Common \: factors \times Remaining \: factor \\ &= ab(a - b) \times (a + b)\\ &= ab(a^2 - b^2).\end{align*}
Lesson
Algebra
Subject
Compulsory Mathematics
Grade
Grade 10
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