ICSE Class 8 Maths Numbers Chapter 02 Factors and Multiples

Factors and Multiples

Factors and Multiples

If a given number is exactly divisible by another number, that number is called a factor of the given number, while the given number is called a multiple of the other number.

Examples: (i) 10 = 2 × 5. So, 2 and 5 are factors of 10. Also, 10 is a multiple of 2 and 5.

(ii) 70 = 2 × 5 × 7. So, 2, 5 and 7 are factors of 70. Also, 70 is a multiple of 2, 5 and 7. Further, 2 × 5, 2 × 7, 5 × 7 and 2 × 5 × 7 are factors of 70.

Note: Every number is a factor of itself and 1 is a factor of every number.

Every number is a multiple of itself and every number is a multiple of 1.

Every number has countless multiples.

Classification of Numbers

Even Numbers

Numbers that leave no remainder on being divided by 2 are called even numbers. In other words, numbers that are multiples of 2 are even numbers. For example, 0, 4, 78, 126, 5132 and 23450 are even numbers.

Note: The even digits are 0, 2, 4, 6 and 8. Any number having an even digit in the units place is an even number.

Odd Numbers

Numbers not exactly divisible by 2 are called odd numbers. In other words, numbers that are not multiple of 2 are called odd numbers. For example, 1, 13, 125, 4137 and 18249 are odd numbers.

Note: The odd digits are 1, 3, 5, 7 and 9. Any number having an odd digit in the units place is an odd number.

Prime Numbers

A number is called a prime number or prime if it has no other factors except 1 and itself. 2, 3, 5, 7, 11, 13, 17, 19 and 23 are some prime numbers. 1 is not considered a prime number. 2 is the only even prime number and it is the smallest prime number.

Composite Numbers

A number that has two or more prime factors is called a composite number. 4, 6, 8, 21, 136 are some composite numbers.

Co-Prime Numbers

Two numbers are called co-prime or relatively prime if they do not have a common factor except 1. For example, 3 and 4; 17 and 21; and 23 and 25 are co-prime numbers.

Twin Primes

Pairs of prime numbers that differ by 2 are called twin primes. 3 and 5; 5 and 7; 11 and 13; and 17 and 19 are some examples of twin primes.

Prime Factorization

Writing a number as a product of prime numbers is called prime factorization.

Examples: (i) 48 = 2 × 2 × 2 × 2 × 3.

(ii) 1260 = 2 × 2 × 3 × 3 × 5 × 7.

Method of Prime Factorization

Divide the number repeatedly by prime numbers 2, 3, 5, 7, ... till the quotient is 1.

Alternatively, break the number into smaller factors till all the factors are prime.

Express 840 as a Product of Primes

840 = 2 × 420

= 2 × 2 × 210

= 2 × 2 × 2 × 105

= 2 × 2 × 2 × 3 × 35

= 2 × 2 × 2 × 3 × 5 × 7

Another method shows 840 divided by 2 equals 420, divided by 2 equals 210, divided by 2 equals 105, divided by 3 equals 35, divided by 5 equals 7, divided by 7 equals 1.

Therefore, 840 = 2 × 2 × 2 × 3 × 5 × 7.

Highest Common Factor (HCF)

The greatest number which is a common factor of two or more numbers is their highest common factor (HCF) or greatest common divisor (GCD).

Example: The factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24.

The factors of 80 = 1, 2, 4, 5, 8, 10, 16, 20, 40, 80.

The common factors of 24 and 80 = 1, 2, 4, 8 and of these, 8 is the greatest number. So, the HCF of 24 and 80 = 8.

Methods of Finding HCF

There are two methods of finding the HCF of two or more numbers.

(i) By prime factorization (ii) By division

To Find HCF by Prime Factorization

Steps 1. Express each number as a product of prime numbers.

2. Find the common factors of the numbers.

3. Find the product of the common factors. This product is the required HCF.

Find the HCF of 252, 315 and 441

252 = 2 × 126 = 2 × 2 × 63 = 2 × 2 × 3 × 21 = 2 × 2 × 3 × 3 × 7.

315 = 3 × 105 = 3 × 3 × 35 = 3 × 3 × 5 × 7.

441 = 3 × 147 = 3 × 3 × 49 = 3 × 3 × 7 × 7.

The common factors of 252, 315 and 441 = 3, 3, 7.

Therefore, the HCF = 3 × 3 × 7 = 63.

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