Class 8 – Number Theory
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Even numbers: All the natural numbers which are divisible by 2 are known as even numbers e.g., 2, 5, 4, 6, 8, 10…. Odd numbers: All the natural numbers which are not divisible by 2 are known as Odd numbers. e.g., 1, 3, 5, 7…
Prime numbers: Expect 1 each natural numbers which is divisible by only 1 and itself is called as prime number e.g., 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31…. etc. Table of prime numbers (1 – 100):
Twin prime: The difference between two primes is ‘2’, then the primes are called Twin primes. Example: CoPrime numbers: Two natural numbers are called CoPrime (or relatively primes) numbers if they have no common factor other than 1 or in other words. The highest common factor i.e., H.C.F. between coprime numbers is 1. e.g., (15, 16), (14, 25), (8, 9), (13, 15) etc. 
Coprimes need not be primes. Composite numbers: a number other than one which is not a prime number is called a composite number. It mean divisible by some other number(s) other than 1 and the number itself e.g., 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26… Consecutive numbers: A series of numbers in which the next number is 1 more than the previous number or the predecessor number is 1 less than successor or justify they can be differed by 1 e.g., 10, 11, 12, or 17, 18, 19, 20, 21, or 717, 718, 719, 720, 721, … etc. Recall few divisible rules. Divisibility by 4: If the number formed by last two digits of the given number is divisible by 4 then the actual number must be divisible by 4 i.e., the last two digits of a number can be 00, 04, 08, 12, 16, 20, 24, 28, 32, …. , 96 

Divisibility by 8: If the number formed by last three digits of the given number is divisible by 8 then the actual number must be divisible by 8 i.e., the last three digits of a number can be 000, 008, 016,024, 032, 040, ……, 096, 104, ….., 992.
e.g.,8537000, 9317640, 39455080, 23456008, 12345728, 3152408 etc.
Divisibility by 16: If the number formed by last four digits of the given number is divisible by 16 then the actual number must be divisible by 16 i.e., the last four digits of a number can be 0000, 0016,0032, 0048, 0064, 0080, 0096, 0112, 0128, 0144, 0160, …, 0960, 0976, …., 0992,…., 1600, …., 9984.
Divisibility by 32, 64, 128, .. can be checked just by checking the last 5, 6, 7, … digits number as in the above.
Divisibility by 7: To check the divisibility of a number by 7 we apply the following method. Take the ones place digit in the given number and double it, then subtract it from the cut off number. If the remainder is 0 or a multiple of 7 then it is divisible by 7. If the number is long continue the same above process till the remainder is a simple number. Let the number be 133.
133 ⇒ 13 –3 × 2 = 13 – 6 = 7
Since 7 is divisible by 7, so the given number 133 will be also divisible by 7.
In the above process 2 is multiplied with the last digit it is negative osculate for 7, which we have earlier discussed.
Example: Check whether 1071 is divisible by 7
Solution:
Step 1: 1071 ⇒ 107– (1 × 2) = 105
Step 2: 105 ⇒ 10 –(5 × 2) = 0
Since 0 is divisible by 7 hence the given number 1071 is also divisible by 7.
Example: Check whether 939715 is divisible by 7.
Solution:
Step 1: 939715⇒93971 – (5 × 2) = 93961
Step 2: 93961⇒ 9396 – (1 × 2) = 9394
Step 3: 9394 ⇒ 939 – (4 × 2) = 931
Step 4: 931 ⇒ 93 – (1 × 2) = 91
Step 5: 91 ⇒ 9 –(1 × 2) = 7
Hence it is divisible by 7.
Divisibility by 13: To check the divisibility of a number by 13 we apply the following method.
Example: Check whether 2366 is divisible by 13.
Solution:
Step 1: 2366 ⇒ 236+ (6 × 4) = 260
[Since, the osculator for 13 is 4]
Step 2: 260 ⇒ 26 +(0 × 4) = 26
Since 26 (or 260) is divisible by 13 hence 2366 is also divisible by 13.
Example: Check whether 377910 is divisible by 13.
Solution:
Step 1: 377910⇒37791 + (0 × 4) = 37791
Step 2: 37791⇒ 3779+ (1 × 4) = 3783
Step 3: 3783 ⇒ 378+ (3 × 4) = 390
Step 4: 390 ⇒ 39 +(0 × 4) = 39
Since 39 is divisible by 13. so 377910 is also divisible by 13.
There are total 25 prime numbers up to 100
There are total 46 prime numbers up to 200.