Factor: one number is said to be a factor when it divides the other number exactly. Thus 3 and 4 are factors of 12
Multiple: one number is said to be a multiple of other number when it is exactly divisible by the other.
Common factor: A common factor of two or more numbers is a number that divides each of them exactly. Thus 4 is a common factor of 12,16,24,72
Common multiple: A common multiple of two or more numbers is a number which is exactly divisible by each of them. Thus 18 is a common multiple of 2,3,6 and 9.
Highest Common Factor (HCF): HCF of two or more given numbers is the greatest number that divides each of them exactly. Thus 5 is the HCF of 25 and 35.
HCF is also called Highest Common Divisor or Greatest Common Divisor.
Methods to find HCF
Method of prime factors: Break the given numbers into prime factors and then find the product of the prime factors common to all the numbers. This product will be the required HCF.
Example: HCF of 20, 35 and 45 is
20 = 2 × 2 × 5
35 = 5 × 7
45 = 3 × 3 × 5
The factor common to all the numbers is 5, hence 5 is the HCF.
Method of Division: Divide the greater number by the smaller number, if there is remainder then divide the divisor by the remainder, if again there is remainder, then again divide the divisor by the next remainder and so on until no remainder is left. The last remainder is the required HCF.
Methods to find LCM
Method of prime factors: Divide the given numbers into their prime factors and then find the product of the highest powers of all the factors that occur in the given numbers, and this product will be the required LCM.
Example: LCM of 4, 8 and 24 is
4 = 2 × 2 = 22
8 = 2 × 2 × 2 = 23
24 = 2 × 2 × 2 × 3 = 23 × 3
The prime factors that occur here are 2 and 3. the highest powers of these prime factors are 23 and 31 respectively.
Therefore the required LCM is 23 × 31 = 24.
Regular method: Write all the given numbers in a line and divide them by a number which will exactly divide at least any two of the numbers. write down the quotients and the undivided numbers in a line below the first. Repeat the process until we get a line of numbers which are prime to each other. The product of all the divisors and the numbers in the last line will be the required LCM.
LCM of 8, 12 and 32 is
Therefore required LCM is 2 × 2 × 2 × 3 × 7 × 4 = 672
LCM of decimals: First find LCM of the given numbers without decimals and then put the decimal in the result after the number of digits which is equal to the minimum digits after the decimal in the given numbers from right to left.
LCM of 2.4, 0.012 and 0.32 is
LCM of 24, 12 and 32 is 96
In the given numbers the minimum digits from right to left is in 2.4 i.e., 1
Therefore the required LCM is 9.6.