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Tips and Tricks: Number System | Quantitative Aptitude (Quant) - CAT PDF Download

Introduction

Number Systems is the most important topic in the quantitative section. It is a very vast topic and a significant number of questions appear in CAT every year from this section.  Learning simple tricks like divisibility rules, HCF and LCM, prime number and remainder theorems can help improve the score drastically.

HCF & LCM

  • HCF * LCM of two numbers = Product of two numbers 
  • The greatest number dividing a, b and c leaving remainders of 
  • The greatest number dividing a, b and c (a<b<c) leaving the same remainder each time is the HCF of (c-b), (c-a), (b-a). 
  • If a number, N, is divisible by X and Y and HCF(X,Y) = 1. Then, N is divisible by X × Y

Prime and Composite Numbers

  • Prime numbers are numbers with only two factors, 1 and the number itself.
  • Composite numbers are numbers with more than 2 factors. Examples are 4, 6, 8, 9. 
  • 0 and 1 are neither composite nor prime. 
  • There are 25 prime numbers less than 100.

Properties of Prime numbers

  • 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 number greater than 2 and 3 can be written in the form of 6k+1 or 6k-1 
  • If a and b are coprime then ab-1 mod b=1

Theorems on Prime numbers

  • Fermat's Theorem: Remainder of a(p-1) when divided by p is 1, where p is a prime.
  • Wilson's Theorem: Remainder when (p-1)! is divided by p is (p-1) where p is a prime
  • Remainder Theorem
  • If a, b, c are the prime factors of N such that N= ap* bq*cr Then the number o f numbers less than N and co-prime to N is ϕ(N)= N (1 - 1/a) (1 - 1/b) (1 - 1/c).
  • This function is known as the Euler's totient function.

Euler's theorem

  • If M & N are coprime to each other than remainder when Mϕ(N) is divided by N is 1.
  • Highest power of n in m! is Tips and Tricks: Number System | Quantitative Aptitude (Quant) - CAT Ex: Highest power of 7 in 100! 
  • Tips and Tricks: Number System | Quantitative Aptitude (Quant) - CAT
  • To find the number of zeroes in n! find the highest power of 5 in n!
  • If all possible permutations of n distinct digits are added together the sum = (n-1)! * (sum of n digits) * (11111... n times)
  • If the number can be represented as N = ap* bq*cr... then number of factors the is (p+1) * (q+1)* (r+1)
  • Sum of the factors = Tips and Tricks: Number System | Quantitative Aptitude (Quant) - CAT
  • If the number of factors are odd then N is a perfect square.
  • If there are n factors, then the number of pairs of factors would be n/2.
  • If N is a perfect square then number of pairs (including the square root) is (n+1)/2
  • If the number can be expressed as N = 2p * aq * b.... where the power of 2 is p and a, b are prime numbers
  • Then the number of even factors of N = p (1+q) (1+r)......
  • The number of odd factors of N = (1+q) (1+r)…
  • Number of positive integral solutions of the equation x2 + y2 = k is given by
  • Total number of factors of k/2 (If k is odd but not a perfect square)
  • Tips and Tricks: Number System | Quantitative Aptitude (Quant) - CAT(If k is odd and a perfect square)
  • Tips and Tricks: Number System | Quantitative Aptitude (Quant) - CAT (If k is even and not a perfect square)
  • Tips and Tricks: Number System | Quantitative Aptitude (Quant) - CAT (If it is even and a perfect square)
  • Number of digits in ab = [ b logm (a) ] + 1 ; where m is the base of the number and [.] denotes greatest integer function.
  • Even number which is not a multiple of 4, can never be expressed as a difference of 2 perfect squares.
  • Sum of first n odd numbers is n2.
  • Sum of first n even numbers is n(n+1)
  • The product of the factors of N is given by Na/2, where a is the number of factors
  • The last two digits of a2, (50 - a)2, (50 + a)2, (100 - a2). . . . . are the same.
  • If the number is written as 210n
  • When n is odd, the last 2 digits are 24.
  • When n is even, the last 2 digits are 76

Divisibility

  • Divisibility by 2: Last digit divisible by 2 
  • Divisibility by 4: Last two digits divisible by 4 
  • Divisibility by 8: Last three digits divisible by 8 
  • Divisibility by 16: Last four digit divisible by 16 
  • Divisibility by 3: Sum of digits divisible by 3 
  • Divisibility by 9: Sum of digits divisible by 9 
  • Divisibility by 27: Sum of blocks of 3 (taken right to left) divisible by 27 
  • Divisibility by 7: Remove the last digit, double it and subtract it from the truncated original number.
  • Check if number is divisible by 7 
  • Divisibility by 11: (sum of odd digits) - (sum of even digits) should be 0 or divisible by 11

Divisibility properties

  • For composite divisors, check if the number is divisible by the factors individually.
  • Hence to check if a number is divisible by 6 it must be divisible by 2 and 3.
  • The equation an - bn is always divisible by a-b. If n is even it is divisible by a + b. If n is odd it is not divisible by a + b.
  • The equation an + bis divisible by a+b if n is odd. If n is even it is not divisible by a+b.
  • Converting from decimal to base b. Let R1 + R2 .. be the remainder left after repeatedly dividing the number with b. Hence, the number in base b is given by ... R2 R1.
  • Converting from base b to decimal - multiply each digit of the number with a power of b starting with the rightmost digit and b 0 . 
  • A decimal number is divisible by b-1 only if the sum of the digits of the number when written in base b are divisible by b-1.

Cyclicity

To find the last digit of a n find the cyclicity of a. For Ex. if a=2, we see that

  • 21 = 2
  • 22 = 4
  • 23 = 8
  • 24 = 16
  • 25 = 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 an, The steps are:

  • Find the cyclicity of a, say it is x 2.
  • Find the remainder when n is divided by x, say remainder r
  • Find a r if r > 0 and ax when r = 0
  • (a + b)(a — b) = (a2 — b2
  • (a + b)2 = (a2 + b2 + 2 ab) 
  • (a — b)2 = (a2 + b2 — 2ab) 
  • (a + b + c)2 = (a2 + b2 + 2(ab + be + ca))
  • (a + b + c)2= (a2 + b2 + 2 (ab + bc + ca)) 
  • (a+ b3) = (a + b) (a2 — ab + b2
  • (a3 + b3) = (a + b) (a2 — ab + b2
  • (a3 + b3) = (a + b) (a2 — ab + b2
  • (a3 + b3 + c3 — 3abc) = (a + b) (a2 + b2 + c2 — ab — bc — ac)
  • When a + b + c = 0, then a3 + b3 + c3 = 3abc
The document Tips and Tricks: Number System | Quantitative Aptitude (Quant) - CAT is a part of the CAT Course Quantitative Aptitude (Quant).
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FAQs on Tips and Tricks: Number System - Quantitative Aptitude (Quant) - CAT

1. What is the difference between HCF and LCM?
Ans. HCF (Highest Common Factor) is the largest factor that divides two or more numbers, while LCM (Least Common Multiple) is the smallest multiple that is divisible by two or more numbers.
2. What are prime and composite numbers?
Ans. Prime numbers are numbers greater than 1 that have only two factors, 1 and itself. Composite numbers, on the other hand, have more than two factors. They can be divided evenly by numbers other than 1 and itself.
3. What are the properties of prime numbers?
Ans. Some properties of prime numbers include: - Prime numbers greater than 2 are always odd. - The only even prime number is 2. - Every prime number can be expressed as a product of prime factors. - There are infinitely many prime numbers.
4. What are the theorems on prime numbers?
Ans. Some theorems on prime numbers include: - Euclid's Theorem states that there are infinitely many prime numbers. - Prime Number Theorem gives an approximation for the number of prime numbers less than a given number. - Wilson's Theorem states that a number is prime if and only if (n-1)! + 1 is divisible by n.
5. What is divisibility?
Ans. Divisibility is the property of one number being divisible by another number without leaving a remainder. For example, a number is divisible by 2 if its units digit is divisible by 2, and a number is divisible by 3 if the sum of its digits is divisible by 3.
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