Types of numbers:
Real Numbers:
- Rational numbers.
- Irrational numbers.
Rational Numbers:
Real numbers that can be represented as a simple fraction (p/q).
Natural numbers : n > 0 ; [1,2,3,4...]
Whole numbers : n >= 0; [0,1,2,3...]
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...}.
Even numbers : n%2 = 0 (Remainder 0)
Odd numbers : n%2 = 1 (Remainder 1)
Prime numbers : Numbers which are divisible by only 1 & itself.
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