Factors and Multiples: If a number x divides another number y exactly, we say that x is a factor of y. Also y is called a multiple of x.
Highest Common Factor (HCF)
The H.C.F. of two or more than two numbers is the greatest number that divides each one of them exactly. There are two methods for determining H.C.F.:
Prime factorization method
Step 1: Express each number as a product of prime factors.
Step 2: HCF is the product of all common prime factors using the least power of each common prime factor
Division Method
Step 1: Write the given numbers in a horizontal line separated by commas.
Step 2: Divide the given numbers by the smallest prime number (write in the left side) which can exactly divide all the given numbers.
Step 3: Write the quotients in a line below the first.
Step 4: Repeat the process until we reach a stage where no common prime factor exists for all the numbers.
Step 5: We can see that the factors mentioned in the left side clearly divides all the numbers exactly and they are common prime factors. Their product is the HCF
Lowest Common Multiple (LCM)
The L.C.M. of two or more than two numbers is the least number which is exactly divisible by each one of the given numbers.
Prime factorization method
Step 1 : Express each number as a product of prime factors.
Step 2 : LCM is The product of highest powers of all prime factors.
Division Method
Step 1: Write the given numbers in a horizontal line separated by commas.
Step 2: Divide the given numbers by the smallest prime number which can exactly divide at least two of the given numbers.
Step 3: Write the quotients and undivided numbers in a line below the first.
Step 4: Repeat the process until we reach a stage where no prime factor is common to any two numbers in the row.
Step 5: LCM is The product of all the divisors and the numbers in the last line.
HCF and LCM of Fractions
LCM of fractions =LCM of Numerators/HCF of Denominators
HCF of fractions =HCF of Numerators/LCM of Denominators
Note:
Product of two numbers = Product of their HCF and LCM.
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