Shortcuts for number divisibility ;
- A number is divisible by 2, if its unit’s digit is any of 0, 2, 4, 6, 8.
- A number is divisible by 3, if the sum of its digits is divisible by 3.
- A number is divisible by 4, if the number formed by the last two digits is divisible by 4.
- A number is divisible by 5, if its unit’s digit is either 0 or 5.
- A number is divisible by 6, if it is divisible by both 2 and 3.
- A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8.
- A number is divisible by 9, if the sum of its digits is divisible by 9.
- A number is divisible by 10, if it ends with 0.
- A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11.
- A number is divisible by 12, if it is divisible by both 4 and 3.
- A number is divisible by 14, if it is divisible by 2 as well as 7.
- Two numbers are said to be co-primes if their H.C.F. is 1. To find if a number, say y is divisible by x, find m and n such that m * n = x and m and n are co-prime numbers. If y is divisible by both m and n then it is divisible by x
Important Formulas of Number System
1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2
(1² + 2² + 3² + ….. + n²) = n ( n + 1 ) (2n + 1)/6
(1³ + 2³ + 3³ + ….. + n³) = (n(n + 1)/2)²
Entirety of first n odd numbers = n²
Entirety of first n even numbers = (n + 1)
All Mathematical Formulas
- (a + b)(a – b) = (a² – b²)
- (a + b)² = (a² + b² + 2ab)
- (a – b)² = (a² + b² – 2ab)
- (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
- (a³ + b³) = (a + b)(a² – ab + b²)
- (a³ – b³) = (a – b)(a² + ab + b²)
- (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)
- when a + b + c = 0, then a³ + b³ + c³ = 3abc
All Types of Number System
Natural Numbers
N = {1,2,3,…}.
Whole Numbers
W = {0,1,2,3,…}
Integers
I = {…,-4,-3,-2,-1,0,1,2,3,4,…}
- Positive Integers I+ = {1,2,3,4,…}
- Negative Integers I- = {-1,-2,-3,-4,…}
- Even Numbers = 2, 4, 6, 8, 10, 12,…etc
Odd Numbers
1, 3, 5, 7, 9, 11, 13, 15, 17, 19,…etc.
Prime Numbers
2, 3, 5, 7, 11, 13, …etc
Composite Numbers
4, 6, 8, 9, etc.
Co-primes
(7, 9),(15, 16)
Rational Numbers
3/5, 7/9, 8/9, 13/15 etc.
Irrational Numbers
√2, √3, √7, etc.
Real Numbers
7/9, 2, 5, , 8/9 etc.
Mathematical Formulas:
- (a + b)(a – b) = (a2 – b2)
- (a + b)2 = (a2 + b2 + 2ab)
- (a – b)2 = (a2 + b2 – 2ab)
- (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
- (a3 + b3) = (a + b)(a2 – ab + b2)
- (a3 – b3) = (a – b)(a2 + ab + b2)
- (a3 + b3 + c3 – 3abc) = (a + b + c)(a2 + b2 + c2 – ab – bc – ac)
- When a + b + c = 0, then a3 + b3 + c3 = 3abc
- (a + b)n = an + (nC1)an-1b + (nC2)an-2b2 + … + (nCn-1)abn-1 + bn