Number System

 Number System



A rational number is a number that can be expressed in the form of p/q where p & q are integers and q≠0.

A prime number is a number that has exactly two factors which are 1 and itself.

1.i) How to find if a number is prime or not?

N is a prime number if it is not divisible by numbers lesser than √N.

Example: 191 is a prime number since it is not divisible by 2, 3, 5, 7, 11, and 13 [numbers less than √191 (≈14)].

Note: Prime numbers will always be in the form (6k±1) where k= 1, 2, 3….But not all (6k±1) will be a prime number

Conversion of a decimal number to fraction:

Example: 6.424242………

Let    x  = 6.424242….

      100x  = 642.424242…..

     (100x – x)= (642.424242…. – 6.424242…..)

       99x    = 636

        x     = 636/99

∴ 6.424242…… = 636/99


  Divisibility Rules

A number is divisible by 2 If the last digit is even.

A number is divisible by 3 If the sum of the digits is divisible by 3.

A number is divisible by 4 If the last two digits of the number divisible by 4.

A number is divisible by 5 If the last digit is a 5 or a 0.

A number is divisible by 6 If the number is divisible by both 3 and 2.

A number is divisible by 7 If the number formed by subtracting twice the last digit with the number formed by; the rest of the digits is divisible by 7. Example: 343. 34-(3×2) = 28 is divisible by 7.

A number is divisible by 8 If the last three digits form a number divisible by 8.

A number is divisible by 9 If the sum of the digits is divisible by 9.

A number is divisible by 10 If the last digit of the number is 0.

A number is divisible by 11 If the difference between the sum of digits in even places and the sum of the digits in odd places is 0 or divisible by 11. Example: 365167484 —(3+5+6+4+4) – (6+1+7+8) = 0 365167484 is divisible by 11.


HCF & LCM

The greatest number that will exactly divide a, b and c is HCF(a, b, c).

The least number which is exactly divisible by a, b and c is LCM(a, b, c).

Example: Find the HCF & LCM of 12 & 18 ?

12 = (2^2 )*(3^1) , 18 = (3^2)*(2^1)

HCF = (2^1)*(3^1)=6 , LCM = (2^2)*(3^2)=36

The rule for Finding out HCF and LCM of Fractions

HCF of two or more fractions is given by HCF of Numerator / LCM of Denominators

LCM of two or more fractions is given by LCM of Numerators / HCF of Denominators

Note: Make sure that you reduce the fractions to their lowest forms before you use these formulae.


FACTORS

The factor of a number is the values that divide the number completely. Example: Factors of 10 are 1, 2, 5, and 10.

A number of factors:

Example: 3600

Step 1: Prime factorize the given number

3600 = 36 x 100 = 6^2 x 10^2 = 2^2 x 3^2 x 2^2 x 5^2 = 2^4 x 3^2 x 5^2

Step 2: Add 1 to the powers and multiply.            

(4+1) x (2+1) x (2+1)  = 5 x 3 x 3  = 45

The number of factors of 3600 is 45.

Sum of factors:

Example: 45

Step 1: Prime factorize the given number

45 = 3^2 x 5^1

Step 2: Split each prime factor as a sum of every distinct factor.

(3^0 + 3^1 + 3^2) x (5^0 + 5^1)

The following result will be the sum of the factors = 78

finding the Number of Zeroes in a Factorial Value

Example: 1142! * 348! * 17!

1142/5 –>Quotient 228. 228/5 –> Quotient 45. 45/5 –> Quotient 9. 9/5–> Quotient 1. total=228 + 45 + 9 + 1= 283 zeroes.

348/5–> Quotient 69. 69/5–> Quotient 13. 13/5–>Quotient 2. total=69 + 13 + 2 = 84 zeroes.

17/5–> Quotient 3 total= 3 zeroes.

Total number of zeroes in the expression

is: 283 + 84 + 3 = 370 zeroes.

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