LCM HCF
LCM and HCF is an easy and important topic for all SSC examinations. Every year, 1-2 questions will be asked from this topic and all the questions can be easily solved by following the right approach within a minimum possible time.
Multiple:
If x divides y exactly, then x is called the multiple of y.
Factor:
If x divides y exactly, then y is one of the factors of x.
Divisibility rules:
1. All the numbers which end with 0,2,4,6,8 are exactly divisible by 2.
2. If the sum of the digits of a number is exactly divisible by 3, then the number is exactly divisible by 3.
3. If the last two digits of a number is exactly divisible by 4, then the number is exactly divisible by 4.
4. All the numbers which end with 0 and 5 are exactly divisible by 5.
5. To find out whether a number is divisible by 7, take the last digit of the number and then double it. Then subtract it from the original number excluding the last digit(doubled digit). Repeat the process until it forms a two-digit number which will be easier to know whether it is a multiple of 7 or not.
6. If the sum of the digits of a number is exactly divisible by 9, then the number is exactly divisible by 9.
7. All the numbers which end with 0 are exactly divisible by 10.
8. To find out whether a number is divisible by 11 or not, take the sum of the alternate digits of the number starting from first from the left and then take the sum of the remaining digits. The difference between both the sums should be divisible by 11.
9. If the last two digits of a number, when subtracted from four times the rest of the number, is exactly divisible by 13, then the original number is exactly divisible by 13.
Least Common Multiple (LCM):
LCM is the least number that is exactly divisible by each of the given numbers.
Highest Common Factor (HCF) or Greatest Common Divisor (GCD):
The highest number which divides each of the given numbers exactly is called the HCF or GCD.
Q) Find the LCM of 7, 14, 16, 28.
Solution:
Method 1:
Given numbers are 7, 14, 16 and 28.
Here, The LCM should be greater than or equal to all the given numbers and also it will be a multiple of the greatest number.
Here, 28 is not the LCM because 28 is not a multiple of 16.
The next multiple of 28 is 56.
56 is also not the LCM because 56 is not a multiple of 16.
Similarly, 84 is also not the LCM.
The next multiple of 28 is 112 which is divisible by all four numbers.
Hence, 112 is the LCM of 7, 14, 16 and 28.
Method 2:
Divide all four numbers with a common prime number so that at least two of the numbers will be exactly divisible by the common prime number. Repeat this process till all the numbers are exactly divisible.
Hence, The LCM is $$7\times2\times2\times4 = 112$$.
Q) Find the HCF of 8, 12 and 20.
Solution:
The factors of 8 are 1, 2, 4 and 8.
The factors of 12 are 1, 2, 3, 4, 6 and 12.
The factors of 20 are 1, 2, 4, 5, 10 and 20.
Here, In all the factors, the common factors are 2 and 4.
Hence, 4 is the Highest Common Factor of 8, 12 and 20.
- The product of two numbers is equal to the product of their LCM and HCF
- The LCM of any given numbers is always exactly divisible by each of the given numbers.
- Each of the given numbers is always a multiple of their HCF.
Q) If the LCM and the HCF of two numbers are 24 and 4 respectively and if one number is 8, then find the other number.
Solution:
Let the other number be 'x'.
Product of two numbers = Product of their LCM and HCF
$$8\times x = 24\times4$$
$$x = 12$$
