Finding Common Ground Class 7 Notes

Finding Common Ground Class 7 Notes

Finding Common Ground Class 7 Notes

1. Highest Common Factor (HCF)

Definition : The largest number that divides two or more numbers without leaving a remainder.

2. Prime Numbers

Definition : Natural numbers greater than 1 with no positive divisors other than 1 and themselves.

Examples : 2, 3, 5, 7, 11, etc.

3. Prime Factorisation

Definition : Expressing a number as the product of its prime factors.

Procedure :

1. Start with the smallest prime number (2).
2. Divide the number by the prime until it no longer divides evenly.
3. Move to the next prime and repeat.
4. Continue until the quotient is 1.

Example :

For 60 :

60 ÷ 2 = 30

30 ÷ 2 = 15

15 ÷ 3 = 5

5 ÷ 5 = 1

Prime Factorisation : 60 = 2² × 3² × 5¹

4. Finding Factors Using Prime Factorisation

Method : Use combinations of the prime factors.

Example : For (60 = 2² × 3¹ × 5¹):

Factors : (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60)

5. Conjecture : A proposed mathematical statement that is believed to be true but has not yet been proven.

Example : The sum of two odd numbers is always even.

6. Lowest Common Multiple (LCM)

Definition : The smallest number that is a multiple of two or more numbers. Example :

To find the LCM of (8) and (6):

Prime Factorisation :

(8 = 23)

(6 = 2 × 3)

Highest Powers :

For (2): (2³)

For (3): (3¹)

LCM Calculation :

LCM = 2³ × 3¹ = 24

7. Finding LCM through Prime Factorisation

Similar to HCF : Take the highest powers of each prime factor.

8. Generalisation

A statement that describes a pattern or property that holds true for all cases.

Example : If the HCF of two numbers is one of the numbers, then one number is a factor of the other.

9. Efficient Procedures for HCF and LCM Relationship :

HCF (a, b) × LCM (a, b) = a × b

Properties Involving HCF and LCM :

The product of two numbers is equal to the product of their HCF and LCM.

CHAPTER AT A GLANCE

1. Lowest Common Multiple (LCM): The LCM of two or more numbers is the smallest number that is a common multiple of those numbers.

2. Prime Factorisation : This is the process of breaking down a number into its prime factors, which are the prime numbers that multiply together to give the original number. For example, the prime factorisation of 12 is 2² × 3.

3. Highest Common Factor (HCF): The HCF of two or more numbers is the largest number that can divide all of them without leaving a remainder. For example, the HCF of 12 and 16 is 4, because 4 is the largest number that can divide both 12 and 16 evenly.

4. Greatest Common Divisor (GCD) : This is another name for the Highest Common Factor (HCF).

5. Factors : Factors of a number are the natural numbers that can divide that number without leaving any remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

6. Divisibility : A number is said to be divisible by another number if it can be divided by that number without leaving a remainder. For example, 15 is divisible by 3 because 15 ÷ 3 = 5 with no remainder.

7. Co-prime Numbers : Two numbers are co¬prime if their only common factor is 1. For example, 8 and 15 are co-prime because they do not share any factors other than 1.

8. Remainder : When one number is divided by another, the remainder is what is left over if the division does not result in a whole number. For example, when 10 is divided by 3, the quotient is 3 and the remainder is 1, because 10 = 3 × 3 + 1

The Greatest of All

Highest Common Factor (HCF)

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest number that divides two or more numbers without leaving a remainder.

Example: To find the HCF of 45 and 75, we first perform the prime factorisation of both numbers :

1. (45 = 3 × 3 × 5 )
2. (75 = 3 × 5 × 5)

Next, weidentify the common prime factors and take the lowest power of each :

1. For (3): the minimum power is (31)
2. For (5): the minimum power is (51)

Thus, the HCF is :

HCF = 3 × 5 = 15

Primes

Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. For example, the first few prime numbers are 2, 3, 5, 7,11, and so on.

Prime Factorisation

Prime factorisation is the process of expressing a number as the product of its prime factors.

Procedure for Prime Factorisation :

1. Start with the smallest prime number (2).
2. Divide the number by the prime number until it no longer divides evenly.
3. Move to the next prime number and repeat the process.
4. Continue until the quotient is 1.

Example: To find the prime factorisation of 60:

1. 60 ÷ 2 = 30
2. 30 ÷ 2 = 15
3. 15 ÷ 3 = 5
4. 5 ÷ 5 = 1

Thus, the prime factorisation of 60 is :

60 = 22 × 31 × 51

Factors of a Number Using Prime Factorisation

Once we have the prime factorisation, we can find all factors of the number by taking different combinations of the prime factors.

Example : For (60 = 2² × 3¹ × 5¹), the factors can be calculated as follows :

(1, 2, 3, 4, 5, 6,10,12, 15, 20, 30, 60)

Conjecture

A conjecture is a mathematical statement that is proposed to be true but has not yet been proven. For example, one might conjecture that the sum of two odd numbers is always even.

Finding the HCF of Numbers Using Prime Factorisation

To find the HCF using prime factorisation, follow these steps:

1. Factor each number into its primes.
2. Identify the common prime factors.
3. Take the lowest power of each common prime factor.

Least, but not Last!

Lowest Common Multiple (LCM)

The Lowest Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers.

Example: To find the LCM of 4 and 6 : .

Prime factorisation:

4 = 2²

6 = 2¹ × 3¹

To find the LCM, take the highest power of each prime:

For (2): 2²

For (3): 3¹ Thus, the LCM is :

LCM = 2² × 3¹ = 12

Finding LCM through Prime Factorisation

The process of finding the LCM through prime factorisation is similar to that of finding the HCF, but instead, we take the highest powers of each prime factor.

Patterns, Properties, and a Pretty Procedure!

Generalisation

A generalisation is, if we observe that the HCF of two numbers is one of the numbers when one number is a factor of the other, we can generalise this observation.

Efficient Procedures for HCF and LCM

There are efficient methods to find both HCF and LCM simultaneously. One such method is using the relationship:

HCF (a, b) × LCM (a, b) = a × b This means that if you know the product of two numbers, you can find one if you have the other.

Property Involving both the HCF and the LCM

The property that the product of two numbers is equal to the product of their HCF and LCM is very useful.

HCF × LCM - Product of two numbers.

For example, if we have numbers (a = 12) and (6 = 18):

1. (HCF (12, 18) = 6)
2. (LCM (12, 18) = 36)

We can verify:

12 × 18 = 216

HCF × LCM = 6 × 36 = 216

This confirms the relationship holds true.