Class 7 – Number Theory
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Factor: If a number can be expressed as product of two or more numbers, then each of the number in the product is called the factor of the given number. Example: 36 = 1 × 36 = 2 × 18 = 3 × 12 = 4 × 9 = 6 × 6 ∴ Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18 and 36.
Multiple: If a number is multiplied by 1, 2, 3 …… then the resulting numbers are called multiples of the given number. Example: Multiples of 6 are 6, 12, 18, 24,……… i) 1 is a factor of every number and is the smallest factor. ii) Every number is a factor of itself and is the largest factor. iii) Every number is a multiple of itself. iv) Every number is a multiple of one. v) The number of multiples of a given number are infinite. 
Even and Odd Numbers: Even Numbers: All natural numbers exactly divisible by 2 (or) all the multiples of 2 are known as even numbers. Thus, 2, 4, 6, 8, 10——— are all even numbers. Odd Numbers: All natural numbers which are not exactly divisible by 2 (or) which are not the multiples of 2 are called odd numbers. Thus 1, 3, 5, 7, 9, 11, ———— are all odd numbers. Two consecutive odd numbers (or) even numbers are differ by 2. Even numbers can be represented as 2n where n is an integer. Odd numbers can be represented as 2n + 1 (where n is an integer).
Results to be numbered about even and odd numbers:


i) Sum of even number of times of odd numbers is even number.
ii) If the product of a certain number of numbers is even, then at least one of them is even.
iii) If the product of a certain number of numbers is odd, then none of them is even.
iv) Sum of first n even natural numbers = n (n +1).
v) Sum of first n odd natural numbers = n^{2}
vi) Sum of first n natural numbers = .
Prime number: Numbers having exactly two different factors namely, 1 and the number itself are known as prime numbers, and it must be a whole number greater than 1.
Example: Prime numbers less than 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.
i) 2 is the only even prime number.
ii) The set of prime numbers are uncountable
iii) 2 and 3 are only two primes which are consecutive numbers.
Composite numbers: Numbers which have more than 2 factors are called composite numbers.
Example: 4,6,8,9,10,12,14,15,16,18 are the first 10 composite numbers.
i) 1 is neither prime nor composite. ii) If x and y are two prime numbers then their product xy will have only 1, x, y and xy as factors. iii) Any whole number greater than 1 is either prime (or) composite.
Twin primes: If the difference of two prime numbers is ‘2’, then the primes are called ‘Twin primes’.
Example: 3, 5; 5, 7; 11, 13 are twin primes.
CoPrimes: If two are more natural numbers which have only ‘1’ is the common factor are called Coprimes.
Example: (2, 3), (3, 4, 5), (4, 9), (8, 13, 15), (16, 25) are coprimes.
i) Coprimes need not be primes themselves.
ii) 3, 5, 7 are called prime triplets. These are the only triplets in primes with difference 2.
Example: State whether the numbers 15 and 28 are twin primes or coprimes?
Solution: Factors of 15 = 1, 3, 5, 15
Factors of 28 = 1, 2, 4, 7, 14, 28
∴ 15 and 28 are not prime numbers.
Hence, they are not twin primes.
But both 15 and 28 are having only ‘1’ as the common factor,
∴ 15 and 28 are coprimes.
Example: Express 42 as sum of two prime numbers.
Solution: 42 = 19 + 23.
Prime factorization: The process of writing a composite number as the product of prime factors is called prime factorization of the given number.
Method of prime factorization of a given number:
Step 1: Divide the given number by the smallest prime factor of the given number.
Step 2: Go on dividing each of the subsequent quotients by the smallest prime factor, till the last quotient is prime.
Step 3: Express the given number as the product of all these factors.
Example: Express each of following as a product of prime factors:
i) 168 ii) 98 Solution:
Divisible properties:
Divisibility by 2: If the given number units place is even (or) having 0, 2, 4, 6 or 8, then the given number is divisible by 2.
Example: Check whether 435672 is divisible by 2 or not?
Solution: 435672
2 is divisible by 2
∴ 435672 is divisible by 2.
Divisibility by 3: If the sum of the digits of the given number is divisible by 3 then the number is divisible by 3.
Example: Check whether 32157 is divisible by 3 or not?
Solution: 32157
Sum of the digits = 3 + 2 + 1 + 5 + 7 = 18
18 is divisible by 3.
∴ 32157 is divisible by 3.
Divisibility by 4: If the number formed by the last two digits of the given number is divisible by 4 then the given number is divisible by 4.
i.e., the last two digits of a number can be 00, 04, 08, 12, 16, 20, 24, 28, 32…96.
Example: Check whether 45984 is divisible by 4 or not?
Solution: 45984
84 is divisible by 4
∴ 45984 is divisible by 4.
Divisibility by 5: If the given number units place ends with ‘0’ or ‘5’ then the number is divisible by 5.
Example: Check whether 25985 is divisible by 5 or not?
Solution: 2515690
5 is divisible by 5
∴ 25985 is divisible by 5.
Divisibility by 6: If the given number is divisible by 2 and 3, then the number is divisible by 6.
Example: Check whether 54336 is divisible by 6 or not?
Solution: 54336
6 is divisible by 2
∴ 54336 is divisible by 2.
Sum of the digits = 5 + 4 + 3 + 3 + 6 = 21
21 is divisible by 3.
54336 is divisible by 3.
54336 is divisible by 2 and 3.
54336 is divisible by 6.
Divisibility by 7: If double the last digit of a number subtracted from the rest of the number and the answer is zero (or) divisible by 7, then the number is divisible by 7.
Example: Check whether 23569 is divisible by 7 or not?
Solution: 23569 (double of 9 is 18)
Difference = 2356 – 18 = 2338
2338 (double of 8 is 16).
Difference = 233 – 16 = 217 is divisible by 7.
∴ 23569 is divisible by 7.
Divisibility by 8: If the given number last three digits are divisible by 8, then the number is divisible by 8.
Example: Check whether 58968 is divisible by 8 or not?
Solution: 58968
968 is divisible by 8
∴ 58968 is divisible by 8.
Divisibility by 9: If the sum of the digits of the given number is divisible by 9 then the number is divisible by 9.
Example: Check whether 75213 is divisible by 9 or not?
Solution: 75213
Sum of the digits = 7 + 5 + 2 + 1 + 3 = 18
18 is divisible by 9.
∴ 75213 is divisible by 9.
Divisibility by 10: If the given number units place ends with ‘0’ then the number is divisible by 10.
Example: Check whether 73290 is divisible by 10 or not?
Solution: 73290
0 is divisible by 10
∴ 73290 is divisible by 10.
Divisibility by 11: If the difference of the sum of the digits of odd places and even places is equal to zero or multiple of 11, then the number is divisible by 11.
Example: Check whether 94572258 is divisible by 11 or not?
Solution:
Sum of the digits of odd places = 8 + 2 + 7 + 4 = 21
Sum of the digits of even places = 5 + 2 + 5 + 9 = 21
Difference = 21 – 21 = 0
‘0’ is divisible by 11
∴ 94572258 is divisible by 11.
Divisibility by 12: If the given number is divisible by 3 and 4, then the number is divisible by 12.
Example: Check whether 78228 is divisible by 12 or not?
Solution: 78228
28 is divisible by 4
∴78228 is divisible by 4.
Sum of the digits = 7 + 8 + 2 + 2 + 8 = 27
27 is divisible by 3.
∴ 78228 is divisible by 3.
∴ 78228 is divisible by 4 and 3.
∴ 78228 is divisible by 12.
If a number is divisible by two or more coprimes, then the number is divisible by the product of them.
Example: 96 is divisible by 3 and 4, then 96 is divisible by 12 (3, 4 are coprimes).
Example: 40 is divisible by 2 and 8, but 40 is not divisible by 16 (2, 8 are not coprimes).
Divisibility by 13: Add four times the last digit to the remaining number. If the result is divisible by 13, then the given number is divisible by 13 (Apply this rule ever and ever again as necessary).
Example: Check whether 50661 is divisible by 13 or not?
Solution: 50661 5066 + 4 (4 times 1 is 4) = 5070
507 + 0 (4 times 0 is 0) = 507 50 + 28 (4 times 7 is 28) = 78
78 is divisible by is 13,
∴ 50661 is divisible by 13.
Divisibility by 16: If the number formed by the last four digits of the given number is divisible by 16 then the given number is divisible by 16.
Example: Check whether 547808 is divisible by 16 or not?
Solution: 547808
7808 is not divisible by 16
∴ 547808 is not divisible by 16.
If a number is exactly divisible by powers of 2 then the number of digits from its units place is verified based on the power.
Example: 1742 is divisible by 2. (2^{1} = 2)
37496 is divisible by 4. (2^{2} = 4)
274984 is divisible by 8. (2^{3} = 8)