Number System

Types of numbers:

Real Numbers:

1. Rational numbers.
2. Irrational numbers.

Rational Numbers:

Real numbers that can be represented as a simple fraction (p/q).

1. Natural numbers : n > 0 ; [1,2,3,4...]

2. Whole numbers : n >= 0; [0,1,2,3...]

3. Integers : Collection of whole numbers and negative numbers bu does not include fractions. Integers are represented by symbol Z. Z = {...-3, -2, -1, 0, 1, 2, 3...}.

4. Even numbers : n%2 = 0 (Remainder 0)

5. Odd numbers : n%2 = 1 (Remainder 1)

6. Prime numbers : Numbers which are divisible by only 1 & itself.

7. Composite numbers : Numbers other than primes and greater than 1. 1 is neither prime nor composite number. 2 is only even prime number.

Irrational numbers:

Irrationals are the real numbers that cannot be represented as a simple fraction (p/q).

Pi : 3.14159265358979...
Euler's number : 2.71828182845904...
Golden ratio : 1.61803398874989...

Divisibility Rules:

2 - If its unit digit is 0, 2, 4, 6 or 8

10, 22, 34, 462

3 - If sum of digits of number is completely divisible by 3.

123, 570, 693, 111

4 - If last 2 digits are divisible by 4.

104, 912, 2016, 30020

5 - If unit digit of number is either 0 or 5.

10, 100, 255, 395

6 - If a it is simultaneously divisible by 2 & 3.

6, 30, 222, 4530, 2224530

8 - If last 3 digits is divisible by 8.

48, 624, 5336

9 - If sum of digits of number is divisible by 9.

9, 36, 459, 1233

10 - If last unit digit of number is 0.

10, 100, 2340, 49560

11 - If difference of sum of digits in odd places and sum of digits in even places is zero or divisible by 11.

764852, 1606, 489830

12 - If number is divisible by 3 & 9.

12, 192, 1476

Division Algorithm

Dividend = (Divisor x Quotient) + Remainder

Standard form:

Any composite number can be written as product of its prime factors and is called Standard form

240 = 2*2*2*2*3*5 = 2^4 * 3^1 * 5^1

Sum of factors of a number in standard form (p^x * q^y * r^z * ...) is

(p^0+p^1+...+p^x) * (q^0 + q^1 + q^2 ... q^y) * (r^0 + r^1 + ... + r^z) * ...

Number of factors/divisors of a number in standard form (p^x * q^y * r^z * ...) is

(x + 1) * (y + 1) * (z + 1) * ...

Q. Sum of divisors of 40 ?

40 = 2^3 * 3^0 * 5^1
According to above formula,
Sum of divisors = (2^0 + 2^1 + 2^2 + 2^3) * (5^0 + 5^1) = 6*15 = 90
Sum of divisors of 40 = 90

Q. Number of divisors of 544 ?

544 = 2^5 * 17^1
According to above formula,
Number of divisors = (5+1) * (1+1)= 12
Number of divisors for 544 = 12

Thankyou