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If two counting (also called natural) numbers are multiplied together, then each one of these numbers is called a factor of the product so obtained. Thus, in the mathematical sentence 7 × 9 = 63, 7 and 9 are factors and 63 is the product. You can also see that 63 ÷ 7 = 9 and 63 ÷ 9 = 7 and there is no remainder in each case.
Therefore, we can also say,
If a counting number divides another number exactly (without any remainder), then the divisor is a factor of the dividend.
Example 1: List all the factors of the following numbers.
(a) 16
(b) 17
(a) 16 = 1 × 16
= 2 × 8
= 4 × 4
∴ The factors of 16 are 1, 2, 4, 8 and 16.
(b) 17 = 1 × 17
∴ The factors of 17 are 1 and 17.
From the above example, it is clear that,
 Every number has at least two factors—1 and the number itself.
 All the factors of a number are either less than or equal to the number.
A prime number is any natural number which has exactly two factors—1 and the number itself.
For example, 13 = 1 × 13.
There are no other factors of 13, so 13 is a prime number.
Every natural number except 1 that has more than two different factors is called a composite number.
Edurev Tips: The number 1 is neither prime nor composite. It is not a prime because it does not have exactly two factors and it is not composite because it does not have more than two factors.
A Greek mathematician, Eratosthenes, devised a method to find out prime numbers between 1 and 100.
The steps used in this method are as follows:
Step 1: Write all the numbers from 1 to 100 in a table as shown.
Step 2: Cross out 1 as it is not a prime number.
Step 3: Circle 2 and cross out multiples of 2.
Step 4: Circle 3 and cross out multiples of 3.
Step 5: Circle 5 and cross out multiples of 5.
Step 6: Circle 7 and cross out multiples of 7.
Circle all the numbers that are not crossed out.
The circled numbers are all the prime numbers between 1 and 100.
How many prime numbers have you found between 1 and 100?
Edurev Tips: This method is called ‘The Sieve of Eratosthenes’ because factors are used to sift out certain numbers which are multiples just as a sieve sifts out chaff from wheat.
Example 2: Tell whether each of the numbers 48 and 23 is composite or prime.
Twin Primes are the prime numbers that differ by 2.
For example, 3 and 5, 5 and 7, 11 and 13, etc. are twin primes.
Two natural numbers are said to be coprimes, If they have only 1 as a common factor.
For example, 2 and 3, 5 and 7, 3 and 4, 4 and 9, etc. are coprimes.
Any composite number can be written as a product of prime numbers. An expression written as a product of its prime factors is called the prime factorization of that number.
Example 3: Find the prime factorization of 120.
Method 1: Factor Tree
Step 1: Write the number to be factorised at the top.
Step 2: Choose any pair of factors as branches. If either of these factors is composite, factorise again.
Step 3: Choose a pair of factors of each composite number and continue the branches for prime factor(s).
Step 4: Keep factorising till you have a row of prime factors.
Conventionally, the prime factors are written in increasing order.
∴ The prime factorisation of 120 = 2 × 2 × 2 × 3 × 5.
Method 2: Continuous Division
Step 1: Start dividing by the smallest prime factor until you cannot divide any more.
Step 2: Continue dividing by the next higher prime factor.
Step 3: Repeat the process till you get 1.∴ The prime factorisation of 120 is 2 × 2 × 2 × 3 × 5.
Edurev Tips: You can have more factor trees for the number 120. Try! But at the end, the prime factors will be the same.
If a number is a factor of two or more whole numbers, it is called a common factor of these numbers.
For example, 4 × 2 = 8 and 4 × 5 = 20. So, 4 is a common factor of 8 and 20.
The Highest Common Factor (HCF) of two or more numbers is the greatest of the common factors. It is the greatest number which divides the given numbers exactly.
You can find the HCF of two or more numbers by different methods, as follows:
Example 4: Find the HCF of 12 and 40.
Step 1: List all the factors of each number.
Factors of 12: , , 3, , 6 and 12
Factors of 40: 5, 8, 10, 20 and 40.
Step 2: Identify common factors.
The common factors are 1, 2 and 4.
Step 3: Identify largest number in the list of common factors.
Out of these common factors, 4 is the largest.
∴ HCF of 12 and 40 is 4.
Example 5: Find the HCF of 20 and 28.
Step 1: List all the factors of each number by either method.
Step 2: Identify the common factors.
Step 3: HCF is the product of common prime factors.
∴ HCF of 20 and 28 = 2 × 2 = 4.
Edurev Tips: You can get the prime factorisation by making factor trees or by continuous division. However, continuous division is the preferred method.
Example 6: Find the HCF of 12, 18 and 42.
Step 1: Divide the numbers by common factors only.
Step 2: Stop dividing when there are no more common factors except 1.
Step 3: Find the product of common factors. 2, 3 and 7 do not have any common factor except 1, so stop division.
∴ HCF of 12, 18 and 42 = 2 × 3 = 6.
Edurev Tips: You can use any of the three methods listed above to find the HCF of two or more numbers.
It is easier to find the HCF of large numbers by division method.
Example 7: Find the HCF of 217 and 686.
Step 1: Divide the greater number by the smaller number.
Step 2: Make the remainder the new divisor and the original divisor, the new dividend. Repeat the process until you get 0 as the remainder.
Step 3: The last divisor is the required HCF. Here, the last divisor is 7. So, the HCF of 217 and 686 is 7.
In case of three numbers, first find the HCF of any two numbers.
Then, take the last divisor of the two numbers and the remaining number and find their HCF.
Example 8: Find the HCF of 275, 525 and 830.
First, we take 275 and 525. Now, find the HCF of 25 and 830.
∴ The HCF of 275, 525 and 830 is 5.
A multiple of a whole number is the product of the number and any counting number.
We can state some multiples of 2 and 11 as:
From the above discussion, we can observe these properties of multiples:
 Every number is a multiple of 1.
 The smallest multiple of any number (other than zero) is the number itself.
 Every multiple of a number is greater than or equal to that number.
 The number of multiples of a given number is infinite.
Example 9: Find the first five multiples of 3.
∴ The first five multiples of 3 are 3, 6, 9, 12 and 15.
If a number is a multiple of two or more numbers, it is called a common multiple of the numbers.
For example, 2 × 9 = 18.
∴ 18 is a common multiple of 2 and 9.
The smallest number (other than zero) that is a multiple of two or more counting numbers is the least common multiple (LCM) of the numbers. It is the least number which is divisible by all the given numbers.
You can find the LCM of two or more numbers by different methods, as follows:
Example 10: Find the LCM of 3, 6 and 12.
Step 1: List some multiples of each number.
The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 27, 30, 33, 39, ...
The multiples of 6 are 6, 12, 18, 30, 42, 48, ...
The multiples of 12 are 12, 48, ...
Step 2: Identify common multiples.
Some common multiples of 3, 6 and 12 are 24, 36, ...
Step 3: Pick up the least common multiple from the list of common multiples.
Out of these common multiples, 24 is the least.
∴ The least common multiple (LCM) of 3, 6 and 12 is 24.
Example 11: Find the LCM of 16 and 24.
Step 1: Find the prime factorisation of each number.
Step 2: Identify the common factors.
Step 3: Multiply the common factors and the extra factors to get the LCM.
∴ LCM of 16 and 24 is 48.
Example 12: Find the LCM of 20, 28 and 40.
Step 1: Divide the numbers by a prime factor common to at least 2 of the given numbers. Bring down the number as such, if it is not completely divisible by the prime factor.
Step 2: Stop dividing when there is no further common factor except 1.
Step 3: Find the product of the numbers in the left column and the last remainders.
∴ LCM of 20, 28 and 40 = 2 × 2 × 2 × 5 × 7 = 280.
Below, we give some such tests:
Below, we give rules to test the divisibility of a number by composite numbers like 4, 6, 8, 12 and 25:
From the above rules, we observe that:
A number is divisible by another number if it is divisible by its coprime factors.
Thus, number divisible by 2 and 5 will also be divisible by 10.
Example 13: Check the divisibility of 19440 by 18.
The coprime factors of 18 are 2 and 9.
Therefore, to check the divisibility of 19440 by 18, we check whether the given number is divisible by 2 and 9 both.
19440: It is an even number and so is divisible by 2.
Sum of the digits = 1 + 9 + 4 + 4 + 0
= 18, which is divisible by 9, so
19440 is also divisible by 9.
Thus, 19440 is divisible by 2 × 9, i.e., 18.
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