Introduction to Prime Numbers
Prime numbers are those numbers that cannot be divided by any other number other than 1 and the number itself without leaving a remainder. This means prime numbers have exactly two divisors the number itself and one.
Examples: 2,3,5,7,11,13,17,19 etc .
To explain this further let us take 17. Number 17 cannot be divided by any other number other than 17 or 1. If we divide 17 by any other number, there will be a reminder or 17 can be expressed only in one form
17 = 17 x 1
Consider number 18. This number 18 can be divided by 1, 2, 3,6 and 9 without leaving a reminder. Hence, 18 is not a prime number. Numbers that are not prime numbers are called composite numbers. 18 is a composite number and can be expressed in many forms like
18 = 1 x 18
18= 2 x 9
18= 3 x 6 etc
Note: Number 1 is considered as neither prime nor composite
Method to Find Prime Numbers
Now we will see how to find prime numbers greater than a particular number. Here, we must note that there is no fixed method of solving this. We need to find out whether the number can be divided by another number other than the number and one. However, a few thumb rules can of help to identify the numbers fast:
Rule 1: All even numbers are composite numbers except 2. If we find a number to be even other than 2 then it cannot be a prime number as it is divided by 2
Rule 2: If the sum of the digits of the number is divisible by 3 then the number itself is divisible by 3 and it is not a prime number. Exception is number 3 itself. Number 3 is a prime number. This rule checks for the divisibility of three.
Rule 3: If the number ends with 5, it is not a prime number. Exception is number 5. Number 5 is a prime number. This rule checks for the divisibility of 5.
Rule 4: If the sum of the odd digits of the number when subtracted from the even digits of the number gives a number divisible by 11 or 0 then the number is divisible by 11 and is not a prime number. Exception is number 11. Number 11 is a prime number.
Ex: Consider number 14641.
Sol: Step 1: The sum of the numbers at odd places is 1+6+1 = 8
Step 2: Sum of the numbers at the even places is 4+4 =8
Step 3: Difference of the sum of the digits at odd and even places is 8-8 =0
Step 4: Hence, the number 14641 is divisible by 11 and is not a prime number.
Let us try some examples
Ex 1: Identify the prime numbers form the list
42, 57,83,75,98
Sol: Step 1: 42 and 98 are even numbers hence are not prime
Step 2: Sum of the digits of 57 is 5+7=12 is divisible by 3.
Hence, the number is divisible by 3 and is not a prime number
Step 3: 75 – Ends with 5 and is divisible by 5 and hence not a prime number
Step 4: 83 – is the prime number in the list
Ex 2: Find the first prime number greater than 109?
Sol: This example can only be solved with a few trial and errors and with intelligent guesses. Let us check some of the numbers after 109. We can eliminate all odd numbers as they cannot be prime numbers.
Step 1: 111- This is divisible by 3 as the sum of the digits is 3
Step 2: 113 – This could be a prime number
Step 3: 115 – This number ends with 5 and not a prime number
Step 4: The maximum possible divisor of a number is half the number. This means that a number can never be divided by a number greater than half its value. So for 113, we need to check whether any number less than 66 can divide the number.
Step 5: 113 cannot be divided by any number divisible by 2, 3 or 5 as 113 itself is not divisible by these numbers. This means that we should check whether 113 is divisible by other prime numbers like 7, 11,13,17,19,23,29,31, 37,41,43,47,53,59 and 61.Since, it is not divisible by any of these numbers, it is a prime number.
Checking whether a number is prime or not goes by the method of elimination. If the divisibility rules are used and if any of the divisors are found then it is definitely not a prime number. But if the number is big and you are not able to identify any divisors using the divisibility rules we CANNOT conclude that it is a prime number. It can have any higher prime number as divisor
For example 588139 is not divisible by 2,3, 5 or 11 etc. But we can write it as below
588139 = 839 x 701 and hence 588139 is NOT a prime number.
Here, 839 and 701 are prime numbers and hence identifying them as divisors manually will be difficult. However, the process can be carried out easily using soft wares.
Between, if you have problem on these topics adding rational expressions with different denominators, please browse expert math related websites for more help on help with math online.
Exercises on Prime Numbers.
Pro 1: Write the first five prime numbers greater than 100.
Ans: 101, 103, 107, 109 and 113
Pro 2: identify the prime number from the following
568, 965, 983, 1353
Ans: 983
Pro 3: Find the first prime number greater than 856
Ans: 857
Prime numbers are those numbers that cannot be divided by any other number other than 1 and the number itself without leaving a remainder. This means prime numbers have exactly two divisors the number itself and one.
Examples: 2,3,5,7,11,13,17,19 etc .
To explain this further let us take 17. Number 17 cannot be divided by any other number other than 17 or 1. If we divide 17 by any other number, there will be a reminder or 17 can be expressed only in one form
17 = 17 x 1
Consider number 18. This number 18 can be divided by 1, 2, 3,6 and 9 without leaving a reminder. Hence, 18 is not a prime number. Numbers that are not prime numbers are called composite numbers. 18 is a composite number and can be expressed in many forms like
18 = 1 x 18
18= 2 x 9
18= 3 x 6 etc
Note: Number 1 is considered as neither prime nor composite
Method to Find Prime Numbers
Now we will see how to find prime numbers greater than a particular number. Here, we must note that there is no fixed method of solving this. We need to find out whether the number can be divided by another number other than the number and one. However, a few thumb rules can of help to identify the numbers fast:
Rule 1: All even numbers are composite numbers except 2. If we find a number to be even other than 2 then it cannot be a prime number as it is divided by 2
Rule 2: If the sum of the digits of the number is divisible by 3 then the number itself is divisible by 3 and it is not a prime number. Exception is number 3 itself. Number 3 is a prime number. This rule checks for the divisibility of three.
Rule 3: If the number ends with 5, it is not a prime number. Exception is number 5. Number 5 is a prime number. This rule checks for the divisibility of 5.
Rule 4: If the sum of the odd digits of the number when subtracted from the even digits of the number gives a number divisible by 11 or 0 then the number is divisible by 11 and is not a prime number. Exception is number 11. Number 11 is a prime number.
Ex: Consider number 14641.
Sol: Step 1: The sum of the numbers at odd places is 1+6+1 = 8
Step 2: Sum of the numbers at the even places is 4+4 =8
Step 3: Difference of the sum of the digits at odd and even places is 8-8 =0
Step 4: Hence, the number 14641 is divisible by 11 and is not a prime number.
Let us try some examples
Ex 1: Identify the prime numbers form the list
42, 57,83,75,98
Sol: Step 1: 42 and 98 are even numbers hence are not prime
Step 2: Sum of the digits of 57 is 5+7=12 is divisible by 3.
Hence, the number is divisible by 3 and is not a prime number
Step 3: 75 – Ends with 5 and is divisible by 5 and hence not a prime number
Step 4: 83 – is the prime number in the list
Ex 2: Find the first prime number greater than 109?
Sol: This example can only be solved with a few trial and errors and with intelligent guesses. Let us check some of the numbers after 109. We can eliminate all odd numbers as they cannot be prime numbers.
Step 1: 111- This is divisible by 3 as the sum of the digits is 3
Step 2: 113 – This could be a prime number
Step 3: 115 – This number ends with 5 and not a prime number
Step 4: The maximum possible divisor of a number is half the number. This means that a number can never be divided by a number greater than half its value. So for 113, we need to check whether any number less than 66 can divide the number.
Step 5: 113 cannot be divided by any number divisible by 2, 3 or 5 as 113 itself is not divisible by these numbers. This means that we should check whether 113 is divisible by other prime numbers like 7, 11,13,17,19,23,29,31, 37,41,43,47,53,59 and 61.Since, it is not divisible by any of these numbers, it is a prime number.
Checking whether a number is prime or not goes by the method of elimination. If the divisibility rules are used and if any of the divisors are found then it is definitely not a prime number. But if the number is big and you are not able to identify any divisors using the divisibility rules we CANNOT conclude that it is a prime number. It can have any higher prime number as divisor
For example 588139 is not divisible by 2,3, 5 or 11 etc. But we can write it as below
588139 = 839 x 701 and hence 588139 is NOT a prime number.
Here, 839 and 701 are prime numbers and hence identifying them as divisors manually will be difficult. However, the process can be carried out easily using soft wares.
Between, if you have problem on these topics adding rational expressions with different denominators, please browse expert math related websites for more help on help with math online.
Exercises on Prime Numbers.
Pro 1: Write the first five prime numbers greater than 100.
Ans: 101, 103, 107, 109 and 113
Pro 2: identify the prime number from the following
568, 965, 983, 1353
Ans: 983
Pro 3: Find the first prime number greater than 856
Ans: 857
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