Real Numbers, HCF & LCM

Real Numbers

The type of number we normally use, such as 1, 15.82, −0.1, 3/4, etc.

Positive or negative, large or small, whole numbers or decimal numbers are all Real Numbers.

They are called “Real Numbers” because they are not Imaginary Numbers.

HCF & LCM (Simple Problems)

LCM

The LCM of a set of two or more numbers is the smallest of their common multiples. Multiples mean the numbers which follow as the result of multiplying the number with numbers like 1,2,3 etc. To find the common multiples all we need to do is see what numbers end up to be the common multiple for all the given numbers.

For example, when we find the LCM of 9 and 12 we need to find the common multiples. The common multiples of 9 are 36,72,108 etc… The smallest of these is 36, hence 36 shall be the LCM of 9 and 12.

LCM by Prime factorization

To find the LCM using prime factorization method we need to follow the below-mentioned steps:

• Find the prime factors of numbers individually.
• From all the factors, identify the maximum number of times each prime factor appears.
• The product of the prime factors occurring in maximum numbers is the LCM of the given set of numbers.

Let us use the steps in the following example: find the LCM of 8 and 24.

Step 1: First find the prime factors of the numbers 8 and 24

• Prime factors of 8 = 2×2×2
• Prime Factors of 24= 2×2×2×3

Step 2: Choose out the number occurring a maximum number of times. The number 2 occurs 3 times and 3 occurs 1 time. number occurring the maximum number of times is 2×2×2×3.

Step 3: The product of these numbers is 24. So the LCM of 8 and 24 is 24.

LCM by Division Method

For calculating the LCM by division method we need to follow the below mentioned steps:

• First, write all the given numbers in a single row but separated by commas.
• Find the least prime number that divides at least two numbers from the set of given numbers.
• Write the quotients exactly below the respective number. The numbers which are not divisible by that prime number have to be written as they are, below the respective number.
•  Keep repeating the step 2 till no two numbers are divisible by the same number.
• To find the LCM, multiply the divisors and remaining quotients. The product of all is the LCM of the given set of numbers.

HCF

The HCF or Highest Common Factor of two or more numbers is the greatest common factor of the given set of numbers. In other words, HCF is the greatest number which exactly divides two or more given numbers.

HCF by Listing Method

The listing method involves the process of listing the factors of the given numbers. For example, find the HCF of 20 and 35.

• All possible factors of 20 are 1,2,4,5,10 and 20
• All possible factors of 60 are 1,3,4,5,6,10,12,15,20,30,60

The common factors of the given numbers are : 1,2,4,5,10,20. The greatest among all other numbers is 20, so it shall be the HCF of both the numbers.

HCF by Prime Factorization

Before finding HCF by prime factorization we need to know the concept of the same. Let’s take a number say, 45. Now the factors of 45 are 1,3,5,9,15 and 45 itself. Now, apart from 3 and 5 the other numbers 9 and 15 are composite numbers. We hence further factorize them with 9= 3×3 and 15=3×5.

So the factors of 45 shall be only 1,3,3, and 5. This is prime factorization. We now define prime factorization as the process of expressing the number as the product of its prime factors. The prime factors include only prime numbers and not composite numbers.

When we find HCF by prime factorization method, we are finding the greatest common factor among the prime factors or numbers. Steps to be followed for the method are:

1. Find the prime factors of each of the given number.
2. Next, we identify the common prime factors of the given numbers
3. We then multiply the common factors. The product of these common factors is the HCF of the given numbers.

Let us use these steps in the example below: find the HCF of  36 and 48.

Step 1: Finding prime factors individually:

• All possible factors of 36 are: 2×2×3×3×1
• All possible factors of 48 are: 2×2×2×2×3×1

Step 2: Choose out the common factors: 2×2×3

Step 3: Multiply all the common factors to get the HCF of the given numbers:

Here the given numbers are 36 and 48. The product of the common factors: 2×2×3 = 12. So the HCF for the numbers 36 and 48 is 12.