All rational and irrational numbers are combined to form real numbers. On the number line, any real number can be represented.

- This lemma is essentially equivalent to :
*dividend = divisor × quotient + remainder*.

- The Fundamental Theorem of Arithmetic states that the prime factorisation for a given number is unique if the arrangement of the prime factors is ignored.
- Example: 36=2×2×3×3 OR, 36=2×3×2×3

The method of representing a natural number as a product of prime numbers is known as prime factorization.

1.Example: To find the Least Common Multiple (**L.C.M**) of 36 and 56 36=2×2×3×3

2.56=2×2×2×7

3. The common prime factors are 2×2

4. The uncommon prime factors are 3×3 for 36 and 2×7 for 56.

5.LCM of 36 and 56 = 2×2×3×3×2×7 which is 504

H.C.F can be found using two methods – Prime factorisation and Euclid’s division algorithm.

Prime Factorisation :

Given ** **two numbers, we express both of them as products of their respective prime factors. Then, we select the prime factors that are common to both the numbers**Example:** To find the H.C.F of 20 and24 20=2×2×5 and 24=2×2×2×3

The Factor common to 20 and 24 is 2×2, which is 4, which in turn is the H.C.F of 20 and 24.

It is the repeated use of Euclid’s division lemma to find the H.C.F of two numbers.

Example: To find the HCF of 18 and 30

**Example : **Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

96 = 25 × 3, 404 = 22 × 101

Therefore, the HCF of these two integers is 22 = 4.

LCM = 96*404/HCF(96,404) = 96*404/4 = 9696.

**Example : **Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

**Solution : **

We have : 6 = 2 × 3, 72 = 23 × 32, 120 = 23 × 3 × 5

Here, 21 and 31 are the smallest powers of the common factors 2 and 3, respectively.

So, HCF (6, 72, 120) = 21 × 31 = 2 × 3 = 6 23, 32 and 51 are the greatest powers of the prime factors 2, 3 and 5 respectively involved in the three numbers.

So, LCM (6, 72, 120) = 23 × 32 × 51 = 360.

**Example:** Find the largest number that divides 70 and 125 leaving the remainder 5 and 8 respectively.

**Solution:** First, subtract the remainder from the number. 70-5 = 65 125-8 = 117.

To find the largest number, take the HCF of 65 and 117.

65 = 5×13 117 = 3×3×13.

Hence, HCF (65, 117) = 13.

•HCF*LCF= one number *another number.

•LCM of fractions = LCM of numerator/HCF of denominator

•HCF of fractions = HCF of numerator/LCM of denominator

Irrational numbers-

Any number that cannot be expressed in the form of p/q (where p and q are integers and q≠0.) is an irrational number. Example – 4.965458.. , 3.14592….

*Terminating *decimals are decimals that end at a certain point. Example: 0.2, 2.56 and so on.*Non-terminating* decimals are decimals where the digits after the decimal point don’t terminate. Example: 0.333333….., 0.13135235343…

Complex numbers-

a complex number is a simple representation of addition of two numbers, i.e., real number and an imaginary number. One part of it is purely real and the other part is purely imaginary.

Integers are made up of both positive and negative numbers. Integers, on the other hand, are numbers that can be positive, negative, or zero but not a fraction.

On integers, we can do all arithmetic operations such as addition, subtraction, multiplication, and division.

Integers include numbers such as 1, 2, 5,8, -9, -12, and so on. Integers are denoted by the letter “Z.”

it is defined as the parts of a whole. The whole can be an object or a group of objects. In real life, when we cut a piece of cake from the whole of it, then the portion is the fraction of the cake.

1.**Proper fraction – **The proper fractions are those where the numerator is less than the denominator. For example, 8/9 will be a proper fraction since “numerator < denominator”.

2.**Improper fraction –** The improper fraction is a fraction where the numerator happens to be greater than the denominator. For example, 9/8 will be an improper fraction since “numerator > denominator”.**3. Mixed fraction –** mixed fraction is a combination of the integer part and a proper fraction. These are also called mixed numbers or mixed numerals. For example:

**4. Compound fraction- **a fraction of a fraction is called the compound fraction.**5. Complex fraction-** any complicated combination of the other type of fractions.

**Prime numbers – **the number that have only two factors 1 and number itself. Eg-2,3,5 etc.

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