Number Systems is a vast and crucial topic in quantitative aptitude. Mastering concepts like divisibility, prime numbers, and remainder theorems through shortcuts can significantly improve your score. This guide covers the fundamentals up to advanced theorems to help you excel.
Natural numbers are the counting numbers starting from 1. They are the most basic numbers used in counting and ordering.
Key Properties:
Examples:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
Note: 0 is not a natural number
Important Formulas:
1. Sum of first n natural numbers = n(n+1)/2
2. Sum of squares of first n natural numbers = n(n+1)(2n+1)/6
3. Sum of cubes of first n natural numbers = [n(n+1)/2]²
Whole numbers are natural numbers including zero. They form the foundation of our number system.
Key Properties:
Examples:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
Note: Whole numbers are the same as natural numbers, just including zero
Important Properties:
1. Identity Property: a + 0 = a and a × 1 = a
2. Commutative Property: a + b = b + a and a × b = b × a
3. Associative Property: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
Integers include all whole numbers and their negative counterparts. They are crucial in representing quantities that can be positive or negative.
Types of Integers:
Examples:
..., -3, -2, -1, 0, 1, 2, 3, ...
Note: Integers extend infinitely in both positive and negative directions
Important Properties:
1. Closure Property: Integers are closed under addition, subtraction, and multiplication
2. Additive Inverse: For any integer a, there exists -a such that a + (-a) = 0
3. Multiplicative Identity: For any integer a, a × 1 = a
Number Line Representation:
Rational numbers are numbers that can be expressed as a ratio of two integers, where the denominator is not zero. They include all integers, fractions, and terminating or repeating decimals.
Key Properties:
Examples:
Important Properties:
1. Closure Property: Rational numbers are closed under addition, subtraction, multiplication, and division (except by zero)
2. Commutative Property: a + b = b + a and a × b = b × a
3. Associative Property: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c)
4. Distributive Property: a × (b + c) = a × b + a × c
Converting Between Forms:
1. Fraction to Decimal: Divide numerator by denominator
2. Terminating Decimal to Fraction: Write as fraction with power of 10 denominator
3. Repeating Decimal to Fraction: Use algebraic method
Example: Converting Repeating Decimal to Fraction
Let x = 0.333...
10x = 3.333...
10x - x = 3.333... - 0.333...
9x = 3
x = 3/9 = 1/3
A number is divisible by another if it can be divided without leaving a remainder. Divisibility rules help us quickly determine if one number is divisible by another without performing the actual division.
Key Concepts:
| Number | Rule | Example | Explanation |
|---|---|---|---|
| 2 | Last digit is even (0,2,4,6,8) | 24, 36, 58 | Any number ending in an even digit is divisible by 2 |
| 3 | Sum of digits divisible by 3 | 123 (1+2+3=6) | If the sum of all digits is divisible by 3, the number is divisible by 3 |
| 4 | Last 2 digits divisible by 4 | 1324 (24÷4=6) | If the last two digits form a number divisible by 4, the entire number is divisible by 4 |
| 5 | Ends with 0 or 5 | 50, 75 | Any number ending in 0 or 5 is divisible by 5 |
| 6 | Divisible by both 2 & 3 | 18, 24 | Must satisfy both divisibility rules for 2 and 3 |
| 7 | Double last digit, subtract from rest | 161 → 16-(2×1)=14 | Double the last digit and subtract it from the rest of the number. If the result is divisible by 7, the original number is divisible by 7 |
| 8 | Last 3 digits divisible by 8 | 2104 (104÷8=13) | If the last three digits form a number divisible by 8, the entire number is divisible by 8 |
| 9 | Sum of digits divisible by 9 | 729 (7+2+9=18) | If the sum of all digits is divisible by 9, the number is divisible by 9 |
| 10 | Ends with 0 | 100, 230 | Any number ending in 0 is divisible by 10 |
| 11 | (Sum of odd digits) - (Sum of even digits) = 0 or divisible by 11 | 121 → (1+1)-2=0 | Subtract the sum of digits in even positions from the sum of digits in odd positions. If the result is 0 or divisible by 11, the number is divisible by 11 |
Detailed Example: Divisibility by 7
Let's check if 161 is divisible by 7:
1. Take the last digit: 1
2. Double it: 1 × 2 = 2
3. Subtract from the rest: 16 - 2 = 14
4. Check if 14 is divisible by 7: Yes (14 ÷ 7 = 2)
Therefore, 161 is divisible by 7
Tips for Using Divisibility Rules:
Rule for 3:
Any number abcd can be written as 1000a + 100b + 10c + d = 999a + 99b + 9c + (a + b + c + d). Each of the numbers other than the bracket is divisible by 3, so for the number to be divisible by 3, the sum of the digits must be divisible by 3.
If not, the remainder when the number is divided by 3 is the same as the remainder when the sum of digits is divided by 3.
Rule for 4:
Any number abcd can be written as 1000a + 100b + 10c + d. 100, 1000, and higher powers of 10 are divisible by 4, so for the number to be divisible by 4, 10c + d (the last two digits) must be divisible by 4.
If not, the remainder when the number is divided by 4 is the same as the remainder when the last two digits are divided by 4.
Rule for 8:
The last three digits should be divisible by 8. If not, the remainder when the number is divided by 8 is the same as the remainder when the last three digits are divided by 8.
Rule for 9:
Similar to 3, but with 9. The sum of digits should be divisible by 9. If not, the remainder when the number is divided by 9 is the same as the remainder when the sum of digits is divided by 9.
Rule for 11:
Add all the alternate digits starting with the unit's digit (U). Add all the remaining alternate digits (T). If the difference U - T is 0 or divisible by 11, the number is divisible by 11. If not, the remainder is U - T (if positive) or (U - T) + 11 (if negative).
To check if a number is divisible by a composite number, check divisibility by its co-prime factors. For example, to check divisibility by 24, check for 3 and 8 (since they are co-prime). But divisibility by 4 and 6 does not guarantee divisibility by 24, since 4 and 6 are not co-prime.
E.g. 36 is divisible by 4 and 6, but not by 24. However, if a number is divisible by 8 and 9, it is divisible by 72 (since 8 and 9 are co-prime).
When using the rule for 11, if the difference U - T is negative, the remainder is not the absolute value, but (U - T) + 11.
E.g. For 59439152, U = 2+1+3+9 = 15, T = 5+9+4+5 = 23. U - T = -8, so remainder is -8 + 11 = 3.
Example 1: How many distinct values can x assume if 28357x4 is divisible by 8?
Rule for 8: Last three digits (7x4) must be divisible by 8. Try x = 0, 4, 8. So, x can be 0, 4, or 8 (3 values).
Example 2: If the number 18601x57y is divisible by 72, find the value of x + y.
Since 72 = 8 × 9 (co-prime), check divisibility by 8 and 9. For 8: 57y must be divisible by 8. 57 mod 8 = 1, so y = 6. For 9: sum of digits = 34 + x, so x = 2. Thus, x + y = 8.
Example 3: If the number 1735x86y4 is divisible by 11, what is the least value that x – y can assume?
Sum of alternate digits: (4+6+x+3+1) = 14+x, (y+8+5+7) = 20+y. For divisibility, difference = 0, 11, 22, ... The least value of x-y is -5 (when y-x=5).
Prime numbers are natural numbers greater than 1 that have exactly two distinct positive divisors: 1 and the number itself.
Properties of Prime Numbers:
First 20 Prime Numbers:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71
Prime factorization is the process of expressing a number as a product of prime numbers. This is unique for every number (Fundamental Theorem of Arithmetic).
Steps for Prime Factorization:
Example: Prime Factorization of 84
84 ÷ 2 = 42
42 ÷ 2 = 21
21 ÷ 3 = 7
7 ÷ 7 = 1
Therefore, 84 = 2² × 3 × 7
The HCF, also known as GCD (Greatest Common Divisor), is the largest number that divides two or more numbers without leaving a remainder.
Methods to Find HCF:
Example: Finding HCF of 48 and 36
Method 1: Prime Factorization
48 = 2⁴ × 3
36 = 2² × 3²
HCF = 2² × 3 = 12
Take the minimum power of common prime factors
Euclid's Algorithm:
For two numbers a and b:
1. Divide a by b
2. If remainder is 0, b is the HCF
3. If remainder is not 0, replace a with b and b with remainder
4. Repeat until remainder is 0
The LCM is the smallest number that is a multiple of two or more numbers.
Methods to Find LCM:
Example: Finding LCM of 12 and 18
Method 1: Prime Factorization
12 = 2² × 3
18 = 2 × 3²
LCM = 2² × 3² = 36
Take the maximum power of all prime factors
Important Relationship:
For any two numbers a and b:
LCM(a,b) × HCF(a,b) = a × b
This relationship is useful for finding LCM when HCF is known and vice versa
Key Formulas for LCM & HCF
Two numbers are co-prime (or relatively prime) if their highest common factor (HCF) is 1. For example, (8, 9) and (15, 22) are co-prime pairs.
Property of Co-primes:
If a number N is divisible by two co-prime numbers X and Y, then N is also divisible by their product, \( X \times Y \).
A prime number is a natural number greater than 1 that has exactly two distinct factors: 1 and itself. Examples include 2, 3, 5, 7, 11. There are 25 prime numbers less than 100.
A composite number is a natural number greater than 1 that has more than two factors. Examples include 4, 6, 8, 9.
Note: The numbers 0 and 1 are neither prime nor composite.
Fermat's Little Theorem
If p is a prime number, then for any integer a not divisible by p, the remainder of \( a^{p-1} \) when divided by p is 1.
\( a^{p-1} \equiv 1 \pmod{p} \)
Wilson's Theorem
For any prime number p, the remainder when \( (p-1)! \) is divided by p is always (p-1).
\( (p-1)! \equiv -1 \equiv (p-1) \pmod{p} \)
Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after reaching a certain value (the modulus).
Key Concepts:
Examples:
17 ≡ 2 (mod 5) because 17 ÷ 5 = 3 remainder 2
23 ≡ 3 (mod 10) because 23 ÷ 10 = 2 remainder 3
100 ≡ 0 (mod 4) because 100 ÷ 4 = 25 remainder 0
This fundamental theorem states that for any integers a and b (b≠0), there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.
Components:
Example:
For a = 17 and b = 5:
17 = 5 × 3 + 2
where q = 3 and r = 2
Note: 0 ≤ 2 < 5, satisfying the condition
Applications:
1. Finding HCF using Euclidean Algorithm
2. Proving divisibility properties
3. Solving Diophantine equations
For a prime number p, (p-1)! ≡ -1 (mod p). This theorem is useful in primality testing and number theory.
Key Points:
Example:
For p = 5:
(5-1)! = 4! = 24
24 ≡ -1 (mod 5) because 24 ÷ 5 = 4 remainder 4
And -1 ≡ 4 (mod 5)
If p is a prime number and a is not divisible by p, then a^(p-1) ≡ 1 (mod p).
Applications:
Example:
For p = 7 and a = 3:
3⁶ = 729
729 ≡ 1 (mod 7) because 729 ÷ 7 = 104 remainder 1
Corollary:
If p is prime and a is not divisible by p, then a^(p-2) is the modular inverse of a modulo p.
That is, a × a^(p-2) ≡ 1 (mod p)
The remainder of \( \frac{a^n + b^n}{a-b} \) is \( 2b^n \) if n is odd, and 0 if n is even.
Euler's totient function, denoted as \( \phi(N) \), counts the number of positive integers up to a given integer N that are relatively prime to N.
If the prime factorization of N is \( N = a^p \cdot b^q \cdot c^r \ldots \), then the value of the totient function is:
\( \phi(N) = N \left(1 - \frac{1}{a}\right) \left(1 - \frac{1}{b}\right) \left(1 - \frac{1}{c}\right) \ldots \)
Euler's Remainder Theorem states that if M and N are co-prime integers, then the remainder when \( M^{\phi(N)} \) is divided by N is 1.
\( M^{\phi(N)} \equiv 1 \pmod{N} \)
Note: If N is a prime number, \( \phi(N) = N-1 \), and Euler's theorem simplifies to Fermat's Little Theorem.
Converting decimal numbers to binary involves repeatedly dividing by 2 and recording the remainders.
Steps:
Example: Convert 13 to binary
13 ÷ 2 = 6 remainder 1
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Reading from bottom to top: 1101
Therefore, 13₁₀ = 1101₂
Converting binary numbers to decimal involves multiplying each digit by its corresponding power of 2 and adding the results.
Steps:
Example: Convert 1101 to decimal
1101₂ = 1×2³ + 1×2² + 0×2¹ + 1×2⁰
= 1×8 + 1×4 + 0×2 + 1×1
= 8 + 4 + 0 + 1
= 13₁₀
The unit digit of a number raised to a power follows a cyclic pattern. Understanding these patterns helps in quickly finding unit digits.
| Number | Cycle | Pattern | Example |
|---|---|---|---|
| 0,1,5,6 | 1 | Always same | 5^n always ends with 5 |
| 4,9 | 2 | 4,6 / 9,1 | 4^1=4, 4^2=16, 4^3=64 |
| 2,3,7,8 | 4 | Repeats every 4 powers | 2^1=2, 2^2=4, 2^3=8, 2^4=16 |
Example: Find unit digit of 3⁴⁵
Cyclicity of 3 is 4
45 mod 4 = 1
Therefore, unit digit = 3¹ = 3
Quick Method:
1. Find the cyclicity of the base number
2. Find the remainder when the power is divided by the cyclicity
3. The unit digit will be the same as the unit digit of the base number raised to this remainder
If a number N can be expressed as a product of its prime factors as \( N = a^p \cdot b^q \cdot c^r \ldots \), then:
If \( N = 2^p \cdot a^q \cdot b^r \ldots \) (where a, b are odd primes):
The highest power of a prime number 'n' that can divide m! (m factorial) is given by Legendre's formula:
\( \left[ \frac{m}{n} \right] + \left[ \frac{m}{n^2} \right] + \left[ \frac{m}{n^3} \right] + \ldots \)
where [x] is the greatest integer function (floor function).
Example: Highest power of 7 in 100! = \( [\frac{100}{7}] + [\frac{100}{49}] = 14 + 2 = 16 \).
Number of Zeroes in a Factorial
To find the number of trailing zeroes in n!, you need to find the highest power of 5 in n!, since zeroes are formed by pairs of 2 and 5, and the number of 5s is always the limiting factor.
An even number that is not a multiple of 4 can never be expressed as the difference of two perfect squares.
The number of positive integral solutions for the equation \( x^2 - y^2 = k \) depends on k:
If all possible permutations of n distinct digits are added together, the sum is:
Sum = \( (n-1)! \times (\text{sum of n digits}) \times (111\ldots \text{n times}) \)
The number of digits in an integer N is given by \( \lfloor \log_{10}(N) \rfloor + 1 \). For large numbers in the form \( a^b \):
Number of digits in \( a^b = \lfloor b \times \log_{10}(a) \rfloor + 1 \)
Sharpen your number system skills with a wide range of questions, from divisibility and remainders to digit cycles, CAT-level puzzles, and more. Each question comes with a detailed answer and explanation.
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