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Questions Based on Concepts

In this section, we will discuss the concepts and questions based upon them.

LCM and HCF

Meaning of LCM
A common multiple is a number that is a multiple of two or more than two numbers. The common multiples of 3 and 4 are 12, 24,…

The least common multiple (LCM) of two numbers is the smallest positive number that is a multiple of both.
 
Multiples of 3  3, 6, 9, 12, 15, 18, 21, 24,…
 
Multiples of 4  4, 8, 12, 16, 20, 24, 28,…

So, the LCM of 3 and 4 will be 12, which is the lowest common multiple of 3 and 4.
First of all, the basic question is what kind of numbers we can use for LCM.

Let us see this through an example— LCM of 10, 20 and 25 is 100. It means that 100 is the lowest number which is divisible by all these three numbers.

But cannot the LCM be (–100)? Since (
100) is lower than 100 and divisible by each of 10, 20 and 25. Or, it can be zero also.

Or, What will be the LCM of (–10) and 20?
 

Will it be (–20) or (–200) or (–2000) or the smallest of all the numbers, i.e., 
−∝
 

Answer to all these questions is very simple. LCM is a concept defined only for positive numbers, be it an integer or a fraction i.e., LCM is not defined for negative numbers
 or zero.
 

Now, we will define a different method of finding out the LCM of two or more than two positive integ
ers.

Process to find out LCM

Step 1 Factorize all the numbers into their prime factors.
 
Step 2 Collect all the distinct factors.
 
Step 3 Raise each factor to its maximum available power and multiply.
 
Example-1
Find LCM of 10, 20, 25.
Solution
Step 1 10 = 21 × 51
            20 = 22 × 51
            25 = 52
Step 2 2, 5
Step 3 22 × 52 = 100
 
The biggest advantage of using this method lies in the fact that we can find out the LCM of any number of numbers in a straight line without using the conventional method. It can be understood in the following way with the previous example.
 
First of all, find out the LCM of 10, 20 = 20, and now the LCM of 20 and 25 = 100 (For this you will have to check which factor of 25 is not present in 20 and then multiply by this factor. Since 25 has 52 and 20 has 51 only, so we will multiply 20 by 5.)
 

Example-2
Find LCM of 35, 45, 55.
Solution
First of all, find out the LCM of 35 and 45.
 
Now 35 = 51 × 71 and 45 = 32 × 51.
 
So, it can be observed here that 35 does not have 32 in it, so we will multiply 35 by 32.
So, LCM of 35 and 45 = 35 × 32. (You can start with 45 also to find out about the missing factors of 35 in 45.)
 
Now, we will find out the LCM of 35 × 32 and 55
55 = 51 × 111
 
Now, 111 is not there with 35 × 32. So, we will multiply 35 × 32 with 111.
 
So, finally the LCM = 35 × 32 × 111 = 3465.
 

Relationship between factors of a number and its LCM We can use the factors of a number in several ways to produce that under as the LCM.

 
Example-3
N = 72. How many sets of two values (ab) are there for which the LCM (ab) is 72?
Solution
N = 72 = 23 × 32
 
Values of a and b have to be chosen in such a way that both 23 and 32 are contained in either a or b.
 
Possible sets are (72 ,1) (72, 2) (72, 3) (72, 4) (72, 6) …(72, 36), (72, 72). These are a total of 12 sets.
 
The other possible sets are (9, 8), (9, 24), (8, 18), (8, 36),…
 
These are a total of six sets.
 
So, the total number of possible sets are 12 + 6 = 18 sets
 
In general, if N = ap × bq × cr ×…, then the set of values (x, y) such that LCM (xy) is N can be summed up as:
 
Number of sets (x, y) =
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Meaning of HCF

Factors are those positive integral values of a number, which can divide that number. HCF, which is known as GCD (Greatest Common Divisor) also, is the highest value which can divide the given numbers.
 
Factors of 20: 1, 2, 4, 5, 10, 20.
 
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
 
So, 10 will be the HCF of 20 and 30.

Process to find out HCF

Step 1 Factorize all the numbers into their prime factors.
 
Step 2 Collect all the common factors.
 
Step 3 Raise each factor to its minimum available power and multiply.
 
Example
Find HCF of 100, 200 and 250.
Solution
Step 1 100 = 22 × 52
          200 = 23 × 52
          250 = 53 × 21
Step 2 2, 5
Step 3 21 × 52 = 50
 
Alternatively, to find out HCF of numbers like 100, 200 and 250, it is required to observe the quantity which can be taken out common from these numbers. By writing these numbers as (100x + 200y + 250z), it can be very easily observed that 50 is common out of these numbers.
 

Summarizing LCM and HCF

It is very essential to understand the mechanism of finding out LCM and HCF. We can simply understand the mechanism to find out lowest common multiple and highest common factor through this example.
 
Example-1
Find out LCM and HCF of 16, 12, 24.
Solution
No. Multiples Factors
16 16, 32, 48, 64, 80, 96, 112, 128,… 1, 2, 4, 8, 16
12 12, 24, 36, 48, 60, 72, 84, 96, 108,… 1, 2, 3, 4, 6, 12
24 24, 48, 72, 96, 120, 144, 168, 192,… 1, 2, 3, 4, 6, 8, 12, 24
Common Multiple Common Factor
48, 96, 144 1, 2, 4
Lowest Common Multiple Highest Common Factor
48 4
 
Standard Formulae
  1. LCM × HCF = Product of two numbers.
    This formula can be applied only in case of two numbers. However, if the numbers are relatively prime to each other (i.e., HCF of numbers = 1), then this formula can be applied for any number of numbers.
  2. LCM of fractions = LCM of numerator of all the fractions/HCF of denominator of fractions.
  3. HCF of fractions = HCF of numerator of all the fractions/LCM of denominator of fractions.
  4. HCF of (sum of two numbers and their LCM) = HCF of numbers.
 
Example-2
HCF of two natural numbers A and B is 120 and their product = 10,00 0. How many set of values of A and B is/are possible?
Solution
HCF (A, B) = 120  120 is a common factor of both the numbers (120 being the HCF). Hence 120 is present in both the numbers. So the minimum product of A and B = 120 × 120 = 14400. Hence no set of A and B are possible satisfying the conditions.
 





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