Number System
Number system is one of the most important topics for all MBA entrance exams and also in Banking Exams. Number system being a vast topic, examiner always has a variety of questions to ask. Number system is not at all difficult but it is sure extensive.
Number system not only forms the basis of most calculations, also judges your logical ability & decision power. In number systems there are hundreds of concepts and variations, along with various logic attached to them, which makes this seemingly easy looking topic most complex in preparation for the examination.
The students while going through these topics should be careful in capturing the concept correctly, as it’s not the speed but the concept that will solve the question here. The correct understanding of concept is the only way to solve complex questions based on this section. Learning simple tricks like divisibility rules, HCF and LCM, prime number and remainder theorems can help improve the score drastically.
You really need to understand the basic classification of numbers. You just can't play the game without knowing the rules. So 'understand' basic concepts really well. Numbers are fun to learn. If you learn the concepts thoroughly you will find that solving aptitude questions on number system is a cake walk for you. There are lot of concepts involved and hence even a simple question might look a bit too complex or trickier to solve.
SOME IMPORTANT TOPICS IN NUMBER SYSTEM
• Classification of numbers (Special Numbers)
• Divisibility Rules
• Remainder System
• Factor Theory
• Factorial and Applications
• Last ‘n’ Digits
• HCF and LCM
• Base System
Special Numbers
Real numbers, Rational numbers, Irrational numbers, Integers, Fractions, Negative numbers, Whole numbers, Natural numbers, Even numbers, Odd numbers, Composite numbers, Prime numbers, Prime Factors in advance Fibonacci numbers.
Divisibility Rules
• A number is divisible by 2 if it is an even number.
• A number is divisible by 3 if the sum of the 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 the unit digit is either 5 or 0.
• A number is divisible by 6 if the number is divisible by both 2 and 3
• A number is divisible by 8 if the number formed by the last three digits is 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 unit digit is 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 should be zero or a multiple of 11
Basic Formulas
1. (a + b) (a - b) = (a2 - b2)
2. (a + b)2 = (a2 + b2 + 2ab)
3. (a - b)2 = (a2 + b2 - 2ab)
4. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
5. (a3 + b3) = (a + b) (a2 - ab + b2)
6. (a3 - b3) = (a – b) (a2 + ab + b2)
7. (a3 + b3 + c3 - 3abc) = (a + b + c) (a2 + b2 + c2 - ab - bc - ac)
8. When a + b + c = 0, then a3 + b3 + c3 = 3abc
Sum of natural numbers from 1 to n
n(n+1) / 2
Sum of squares of first n natural numbers is
n(n+1) (2n+1) / 6
Sum of the squares of first n even natural numbers is
(2/3) n (n+1) (2n+1)
Properties of Prime Numbers
• Prime numbers are numbers with only two factors, 1 and the number itself.
• To check if n is a prime number, list all prime factors less than or equal to √n. If none of the prime factors can divide n then n is a prime number.
• For any integer a and prime number p, ap−a is always divisible by p
• All prime numbers greater than 2 and 3 can be written in the form of 6k+1 or 6k-1
• If a and b are co-prime then a(b-1) mod b = 1.
• Remainder of a^(p-1) when divided by p is 1, where p is a prime.
• Remainder when (p-1)! is divided by p is (p-1) where p is a prime
Cyclicity
To find the last digit of a find the cyclicity of a. For Ex. if a=2, we see that
1. 2^1=2
2. 2^2=4
3. 2^3=8
4. 2^4=16
5. 2^5=32
Hence, the last digit of 2 repeats after every 4th power. Hence cyclicity of 2 = 4. Hence if we have to find the last digit of a^n, The steps are below:
1. Find the cyclicity of a, say it is x
2. Find the remainder when n is divided by x, say remainder r
3. Find a^r if r>0 and ax when r=0
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