# How to Ace Number System Questions for CAT with Free PDF of Tips and Tricks

## Cat Number System Tricks PDF Free Ayuda Hijacker Engin

## Introduction

If you are preparing for CAT, one of the most important topics that you need to master is number system. Number system is a vast topic that covers various concepts and formulas related to properties of numbers, factorization, HCF and LCM, remainders, base system, unit digit, last two digits, factorials and more. Number system questions can be tricky and time-consuming if you don't know the right methods and shortcuts.

## Cat Number System Tricks Pdf Free ayuda hijacker engin

Fortunately, there are some simple tricks and shortcuts that can help you solve number system questions quickly and accurately. These tricks can save you a lot of time and effort in the CAT exam. In this article, we will share with you some of these tricks and shortcuts that can make number system easy for you.

But before we dive into the tricks, we have a special offer for you. We have prepared a free PDF with all the number system formulas and concepts that you need to know for CAT. This PDF contains best shortcuts that will help you perform better in the quantitative section. You can download this PDF by clicking on the link below.

Download Number Systems Formulas For CAT PDF

## Properties of Numbers

The first thing that you need to know about number system is the basic properties of numbers. There are three types of numbers that you will encounter in CAT: integers, fractions and decimals. Integers are whole numbers that can be positive or negative. Fractions are numbers that can be expressed as a ratio of two integers. Decimals are numbers that have a decimal point.

Some of the basic properties of numbers that you should know are:

Even numbers are divisible by 2, while odd numbers are not.

Prime numbers are numbers that have exactly two factors: 1 and themselves. Composite numbers are numbers that have more than two factors.

A perfect square is a number that can be expressed as the square of an integer. A perfect cube is a number that can be expressed as the cube of an integer.

A palindrome is a number that remains the same when its digits are reversed. For example, 121, 454, 9009 are palindromes.

An armstrong number is a number that is equal to the sum of its digits raised to the power of the number of digits. For example, 153, 370, 371 are armstrong numbers.

Another important concept that you need to know is the divisibility rules for different numbers. These rules can help you determine whether a number is divisible by another number without actually performing the division. Some of the common divisibility rules are:

NumberDivisibility Rule

2A number is divisible by 2 if its unit digit is even.

3A number is divisible by 3 if the sum of its digits is divisible by 3.

4A number is divisible by 4 if its last two digits are divisible by 4.

5A number is divisible by 5 if its unit digit is either 0 or 5.

6A number is divisible by 6 if it is divisible by both 2 and 3.

8A number is divisible by 8 if its last three digits are divisible by 8.

9A number is divisible by 9 if the sum of its digits is divisible by 9.

10A number is divisible by 10 if its unit digit is 0.

11A number is divisible by 11 if the difference between the sum of its odd-positioned digits and the sum of its even-positioned digits is either 0 or a multiple of 11.

12A number is divisible by 12 if it is divisible by both 3 and 4.

## Factorization and Factors

The next concept that you need to know about number system is factorization and factors. Factorization is the process of breaking down a number into its prime factors. For example, the prime factorization of 60 is 2 x 2 x 3 x 5. Factors are the numbers that divide a given number exactly. For example, the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

Some of the important concepts and formulas related to factorization and factors are:

The prime factorization of a number is unique, except for the order of the factors.

The product of two numbers is equal to the product of their prime factors. For example, (2 x 3 x 5) x (2 x 2 x 7) = (2 x 2 x 2 x 3 x 5 x 7).

The LCM (Least Common Multiple) of two or more numbers is the smallest positive number that is divisible by all of them. The LCM can be found by multiplying the highest power of each prime factor present in the numbers. For example, LCM(12,15) = (2^2) x (3^1) x (5^1) = 60.

The HCF (Highest Common Factor) or GCD (Greatest Common Divisor) of two or more numbers is the largest positive number that divides all of them. The HCF can be found by multiplying the lowest power of each prime factor common to the numbers. For example, HCF(12,15) = (2^1) x (3^1) = 6.

The number of factors of a number can be found by adding one to each power in its prime factorization and multiplying them. For example, the number of factors of 60 = (2^2) x (3^1) x (5^1) = (2+1) x (1+1) x (1+1) = 12.

## The sum of factors of a number can be found by multiplying each term in its prime factorization with one more than itself and dividing by one less than itself. For example, the sum of factors of HCF and LCM

Another important concept that you need to know about number system is HCF and LCM. HCF stands for Highest Common Factor or Greatest Common Divisor (GCD), and LCM stands for Least Common Multiple. These concepts are useful for solving problems involving fractions, ratios, proportions, divisibility and more.

Some of the important concepts and formulas related to HCF and LCM are:

The HCF of two or more numbers is the largest positive number that divides all of them. The HCF can be found by multiplying the lowest power of each prime factor common to the numbers. For example, HCF(12,15) = (2^1) x (3^1) = 6.

The LCM of two or more numbers is the smallest positive number that is divisible by all of them. The LCM can be found by multiplying the highest power of each prime factor present in the numbers. For example, LCM(12,15) = (2^2) x (3^1) x (5^1) = 60.

The product of two numbers is equal to the product of their HCF and LCM. For example, 12 x 15 = 6 x 60.

The HCF and LCM of fractions are found by taking the HCF and LCM of their numerators and denominators respectively. For example, HCF(2/3, 4/9) = HCF(2,4)/LCM(3,9) = 2/9 and LCM(2/3, 4/9) = LCM(2,4)/HCF(3,9) = 4/3.

The HCF and LCM of co-prime numbers are 1 and their product respectively. Co-prime numbers are numbers that have no common factor other than 1. For example, 5 and 7 are co-prime numbers, so HCF(5,7) = 1 and LCM(5,7) = 35.

Some of the common methods to find the HCF and LCM of two or more numbers are:

Prime factorization method: This method involves finding the prime factorization of each number and then using the formulas mentioned above to find the HCF and LCM.

Division method: This method involves dividing the larger number by the smaller number and then dividing the divisor by the remainder until the remainder becomes zero. The last divisor is the HCF of the two numbers. The LCM can be found by dividing the product of the two numbers by their HCF.

Euclid's algorithm: This method is similar to the division method, but it uses subtraction instead of division. It involves subtracting the smaller number from the larger number repeatedly until one of them becomes zero. The other number is the HCF of the two numbers. The LCM can be found by dividing the product of the two numbers by their HCF.

## Remainders

The next concept that you need to know about number system is remainders. Remainders are what is left over when one number is divided by another number. For example, when 17 is divided by 5, the quotient is 3 and the remainder is 2. Remainders are useful for solving problems involving divisibility, congruence, modular arithmetic and more.

Some of the important concepts and theorems related to remainders are:

The remainder when a number is divided by another number is always less than or equal to the divisor. For example, when 17 is divided by 5, the remainder is 2 which is less than 5.

The remainder when a number is divided by another number is unique. For example, when 17 is divided by 5, there is only one possible remainder which is 2.

The remainder when a number is divided by another number can be negative or positive. For example, when -17 is divided by 5, the quotient is -4 and the remainder can be either -2 or +3.

The remainder when a number is divided by another number follows some basic rules such as:

The remainder when a sum or difference of two numbers is divided by another number is equal to the sum or difference of their remainders when divided by the same number. For example, when 17 + 13 is divided by 5, the remainder is 0 which is equal to the sum of the remainders when 17 and 13 are divided by 5, which are 2 and 3 respectively.

The remainder when a product of two numbers is divided by another number is equal to the product of their remainders when divided by the same number. For example, when 17 x 13 is divided by 5, the remainder is 1 which is equal to the product of the remainders when 17 and 13 are divided by 5, which are 2 and 3 respectively.

The remainder when a power of a number is divided by another number is equal to the power of its remainder when divided by the same number. For example, when 17^ 3 is divided by 5, the remainder is 3 which is equal to the power of the remainder when 17 is divided by 5, which is 2^ 3.

Fermat's theorem: This theorem states that if p is a prime number and a is any integer that is not divisible by p, then a^(p-1) leaves a remainder of 1 when divided by p. For example, if p = 7 and a = 3, then 3^(7-1) = 729 leaves a remainder of 1 when divided by 7.

Wilson's theorem: This theorem states that if p is a prime number, then (p-1)! leaves a remainder of -1 or p-1 when divided by p. For example, if p = 7, then (7-1)! = 720 leaves a remainder of -1 or 6 when divided by 7.

Euler's theorem: This theorem states that if a and n are co-prime numbers, then a^(phi(n)) leaves a remainder of 1 when divided by n, where phi(n) is the Euler's totient function that gives the number of positive integers less than or equal to n that are co-prime to n. For example, if a = 3 and n = 10, then phi(10) = 4 and 3^(4) = 81 leaves a remainder of 1 when divided by 10.

Chinese remainder theorem: This theorem states that if n1, n2, ..., nk are co-prime numbers and r1, r2, ..., rk are any integers, then there exists a unique integer x such that x leaves a remainder of ri when divided by ni for i = 1,2,...,k. For example, if n1 = 3, n2 = 4, n3 = 5 and r1 = 2, r2 = 3, r3 = 4, then there exists a unique integer x such that x leaves a remainder of 2 when divided by 3, a remainder of 3 when divided by 4, and a remainder of 4 when divided by 5. The value of x can be found by using the formula: x = (r1*M1*m1 + r2*M2*m2 + ... + rk*Mk*mk) mod N where N = n1*n2*...*nk, Mi = N/ni, and mi is the multiplicative inverse of Mi modulo ni. In this case, N = 3*4*5 = 60, M1 = N/3 = 20, M2 = N/4 = 15, M3 = N/5 = 12, n1) mod n1= -8 + (0) mod n1= -8 mod n1= -8 + (n1 - (-8)) mod n1= -8 + (11) mod n1= 3, m2 = M2^-1 mod n2 = 3 mod n2 = 3, m3 = M3^-1 mod n3 = 3 mod n3 = 3. Therefore, x = (r1*M1*m1 + r2*M2*m2 + r3*M3*m3) mod N = (2*20*3 + 3*15*3 + 4*12*3) mod 60 = (120 + 135 + 144) mod 60 = 399 mod 60 = 39. Hence, x = 39 is the unique solution.

## Base System

The next concept that you need to know about number system is base system. Base system is a way of representing numbers using different symbols and place values. The most common base system that we use is the decimal system, which uses 10 symbols (0,1,2,...,9) and has a place value of 10. For example, the number 123 in decimal system means 1 x 10^ 2 + 2 x 10^ 1 + 3 x 10^ 0.

However, there are other base systems that can be used to represent numbers, such as binary system (base 2), octal system (base 8), hexadecimal system (base 16) and so on. These systems use different symbols and have different place values. For example, the number 101 in binary system means 1 x 2^ 2 + 0 x 2^ 1 + 1 x 2^ 0 = 5 in decimal system.

Some of the important concepts and formulas related to base system are:

To convert a number from one base system to another base system, we can use the following methods:

Division method: This method involves dividing the number by the new base repeatedly and taking the remainders as the digits of the new number. For example, to convert 123 from decimal to binary, we can do: 123 / 2 = 61 remainder 1 61 / 2 = 30 remainder 1 30 / 2 = 15 remainder 0 15 / 2 = 7 remainder 1 7 / 2 = 3 remainder 1 3 / 2 = 1 remainder 1 1. The remainders from bottom to top give the binary digits of the number, which are 1111011. Hence, 123 in decimal is 1111011 in binary.

Subtraction method: This method involves subtracting the largest power of the new base that is less than or equal to the number and repeating the process until the number becomes zero. The powers of the new base that are subtracted give the digits of the new number. For example, to convert 123 from decimal to binary, we can do: 123 - 64 (2^6) = 59 (1 in 2^6 place) 59 - 32 (2^5) = 27 (1 in 2^5 place) 27 - 16 (2^4) = 11 (1 in 2^4 place) 11 - 8 (2^3) = 3 (1 in 2^3 place) 3 - 2 (2^2) = 1 (1 in 2^2 place) 1 - 1 (2^1) = 0 (1 in 2^1 place) 0 - 0 (2^0) = 0 (0 in 2^0 place) The powers of 2 that are subtracted from left to right give the binary digits of the number, which are 1111011. Hence, 123 in decimal is 1111011 in binary.

Expansion method: This method involves expanding the number in terms of its place values and then converting each digit to the new base. For example, to convert 123 from decimal to binary, we can do: 123 = 1 x 10^ 2 + 2 x 10^ 1 + 3 x 10^ 0 = (1 x 10 + 0) x 10 + (2 x 10 + 0) x 10 + (3 x 10 + 0) = ((1 x 10 + 0) x 10 + (2 x 10 + 0)) x 10 + (3 x 10 + 0) = (((1 x 10 + 0) x 10 + (2 x 10 + 0)) x 10 + (3 x 10 + 0)) x 10 Now, we can convert each digit to binary by using the division method or any other method. For example, 1 = 0001 2 = 0010 3 = 0011 Then, we can replace each digit with its binary equivalent and get: 123 = (((0001 x 10 + 0000) x 10 + (0010 x 10 + 0000)) x 10 + (0011 x 10 + 0000)) x 10 = (((00010000) x 10 + (00100000)) x 10 = (((00010000 + 00100000) x 10 + (00110000)) x 10 = ((00110000) x 10 + (00110000)) x 10 = (0011000000 + 00110000) x 10 = 0011110000 x 10 = 00111100000 Hence, 123 in decimal is 00111100000 in binary.

To perform arithmetic operations in different base systems, we can use the following methods:

Addition and subtraction: These operations can be performed by using the same rules as in decimal system, but with the new base. For example, to add 101 and 11 in binary system, we can do: 101 + 11 ------ 1000 We start from the rightmost digits and add them. If the sum is less than the base, we write it as it is. If the sum is equal to or greater than the base, we write the remainder and carry over the quotient to the next place. For example, 1 + 1 = 2 in binary system, which is equal to the base. So we write 0 and carry over 1 to the next place. Similarly, to subtract 101 from 1000 in binary system, we can do: 1000 - 101 ------- 111 We start from the rightmost digits and subtract them. If the difference is positive or zero, we write it as it is. If the difference is negative, we borrow one from the next place and add it to the base. For example, 0 - 1 = -1 in binary system, which is negative. So we borrow one from the next place and add it to the base, which is 2. Then we have 2 - 1 = 1 in binary system, which we write as it is.

Multiplication and division: These operations can be performed by using the same rules as in decimal system, but with the new base. For example, to multiply 101 by 11 in binary system, we can do: 101 x 11 ------- 101 1010 ------- 11111 We start from the rightmost digit of the multiplier and multiply it with each digit of the multiplicand. If the product is less than the base, we write it as it is. If the product is equal to or greater than the base, we write the remainder and carry over the quotient to the next place. For example, 1 x 1 = 1 in binary system, which we write as it is. 1 x 0 = 0 in binary system, which we write as it is. Then we shift one place to the left and repeat the process with the next digit of the multiplier. We add all the partial products to get the final product. Similarly, to divide 11111 by 101 in binary system, we can do: __110__ / \ 101 )11111(111 -101 ---- 1010 -1010 ----- 1 the dividend or the remainder becomes zero. The final quotient is 110 and the final remainder is 1 in binary system.

To solve problems involving base systems, we can use the following methods:

Conversion method: This method involves converting the numbers from different base systems to a common base system, usually decimal system, and then performing the required operation. For example, to find the value of (101)_2 + (11)_8 - (10)_16 in decimal system, we can do: (101)_2 = 1 x 2^ 2 + 0 x 2^ 1 + 1 x 2^ 0 = 4 + 0 + 1 = 5 (11)_8 = 1 x 8^ 1 + 1 x 8^ 0 = 8 + 1 = 9 (10)_16 = 1 x 16^ 1 + 0 x 16^ 0 = 16 + 0 = 16 Therefore, (101)_2 + (11)_8 - (10)_16 = 5 + 9 - 16 = -2 in decimal system.

Direct method: This method involves performing the required operation directly in the given base system without converting to any other base system. For example, to find the value of (101)_2 + (11)_8 - (10)_16 in binary system, we can do: (101)_2 + (11)_8 - (10)_16 = (101)_2 + (1001)_2 - (10000)_2 = (1110)_2 - (10000)_2 = (-1010)_2 in binary system.

## Unit Digit and Last Two Digits

The next concept that you need to know about number system is unit digit and last two digits. Unit digit is the rightmost digit of a number. Last two digits are the rightmost two digits of a number. For example, the unit digit of 123 is 3 and the last two digits of 123 are 23. Unit digit and last two digits are useful for solving problems involving divisibility, cyclicity, patterns and more.

Some of the important concepts and formulas related to unit digit and last two digits are:

The unit digit and last two digits of a number depend on its power and not on its base. For example, the unit digit and last two digits of 3^ 4 are the same in any base system.

The unit digit and last two digits of a number follow a cyclical pattern. For example, the unit digit of powers of 3 are: 3, 9, 7, 1, 3, 9, ... and the last two digits of powers of 3 are: 03, 09, 27, 81, 43, ... These patterns repeat after a certain interval called the cycle length. The cycle length depends on the base and the number. For example, the cycle length of unit digit of powers of 3 in decimal system is 4.

The unit digit and last two digits of a sum or difference of two numbers are equal to the sum or difference of their unit digit and last two digits respectively. For example, the unit digit and last two digits of 123 + 456 are equal to the unit digit and last two digits of 3 + 6 and 23 + 56 respe