## Chapter 1: SET THEORY

### Introduction

Set: A well defined collection of objects is called a set.
Finite set: a set of finite elements.
Null set:  set with no elements.
Singleton set: set with one element only.
Equal set: each element of one set is in the other and vice versa.
Subset: Every element of A is element of B. Denote it as A⊆B A={1,2} and B={1,2,3} or B={1,2}. Proper subset is if there is at least one element in B but not in A but all elements in A are in B. A⊂B. E.g: A = {1,2,3} and B={1,2,3,4}.
Union of sets:
A∪B = {all elements of A and B but no duplicates};
Intersection of sets: A∩B = {common elements of A and B but no duplicates}
Difference of sets: A-B = {set of all elements of A not in B}
Complement of set: If A is the subset of X then the a set of all elements in X but not in A is Ac

Note: A set of 'n' elements can have total 2n
subsets.

### Laws of Indices:

(i) am * an = am + n

(ii) am/an = am - n

(iii) (am)n = am * n

(v) a-n = 1/an

(vi) n$\sqrt{a}$m = am/n

(vii) (ab)m= am ∙ bm.

(viii) (a/b)m = am/bm

### Equations

• A∪B = n(A) + n(B) - n(A ∩B) and
if A ∩B = not empty then n(A-B) + n(B-A) + n(A ∩B)
• n(A∪B∪C) = n(A) + n(B) + n(C) - n(A ∩B) - n(B ∩C) -n(A ∩C) + n(A ∩B ∩C)

For the below Venn diagram if you want to find the formula for any region you can use simple addition and subtraction.

Sample questions that can be asked are:
A = People who love football, B = people who love basketball, C = people who love carrom, U = all people. Then find out
1. people who love only football
2. people who love only carrom
3. people who love only basketball
4. people who love carrom and basketball
5. people who love carrom and football
6. people who love football and basketball.
7. people who love all three.
8. people who love none.
9. Region that represents any of the above. In such cases the regions shall have numbers and not equations.

### Number System

Divisibility rules:

1. Divisible by 2: If its digit in units place is divisible by 2. E.g: 578 is divisible by 2 as units digit = 8 is divisible.
2. Divisible by 3: If sum of digits is divisible by 3. E.g: 372 = 3+7+2 = 12 which is divisible by 3 so 372 is also divisible.
3. Divisible by 4: If the number formed by the last two digits is divisible by 4. E.g: 340 is divisible as 40 is divisible.
4. Divisible by 5: If units digits is 5 or 0. E.g. 55 / 70.
5. Divisible by 6: If the number is divisible by both 2 and 3.
6. Divisible by 8: Number formed by last three digits is divisible by 8.
7. Divisible by 9: Sum of the digits is divisible by 9.
8. Divisible by 10: Last digit should be 0.
9. Divisible by 11: Difference between sum of digits at odd place and even place is 0 or divisible by 11. E.g: 121 i.e. 1+1 - 2 = 0 so divisible.
10. Divisible by 12: A number is divisible by 3 and 4.
11. Divisible by 14: A number is divisible by 2 and 7.
12. Divisible by 15: A number is divisible by both 3 and 5.
13. Divisible by 16: Last four digits form a number divisible by 16.
14. Divisibility By 24 : A given number is divisible by 24, if it is divisible by both 3 and 8.

15. Divisibility By 40 : A given number is divisible by 40, if it is divisible by both 5 and 8.

16. Divisibility By 80 : A given number is divisible by 80, if it is divisible by both 5 and 16.

Important Notes :

• If a number is divisible by "p" as well as "q", where "p" and "q" are co-primes, then the given number is divisible by "pq".

• If p and q are not co-primes, then the given number need not be divisible by pq, even when it is divisible by both p and q.

• Example :36 is divisible by both 4 and 6, but it is not divisible by (4x6) = 24, since 4 and 6 are not co-primes.

Ans .

yes

1. Explanation :

Sum of digits in 541326 = (5 + 4 + 1 + 3 + 2 + 6) = 21, which is divisible by 3. Hence, 541326 is divisible by 3

Ans .

no

1. Explanation :

Sum of digits in 5967013 = (5 + 9 + 6 + 7 + 0 + 1 + 3) = 31, which is not divisible by 3. Hence, 5967013 is not divisible by 3.

• (a + b)2 = a2 + b2 + 2ab

• (a - b)2 = a2 + b2 - 2ab

• (a + b)2 - (a - b)2 = 4ab

• (a + b)2 + (a + b)2 = 2 (a2 + b2)

• (a2 - b2) = (a + b) (a - b)

• (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)

• If a + b + c = 0, then a3 + b3 + c3 = 3abc

• If we divide a given number by another number, then : Dividend = (Divisor x Quotient) + Remainder

1. (xn - an) is divisible by (x - a) for all values of "n"

2. (xn - an) is divisible by (x + a) for even values of "n"

3. (xn + an) is divisible by (x + a) for odd values of "n"

Some Properties of Prime Numbers

• The lowest prime number is 2. 2 is also the only even prime number. The lowest odd prime number is 3.

• The remainder when a prime number p ≥ 5 is divided by 6 is 1 or 5. However, if a number on being divided by 6 gives a remainder of 1 or 5 the number need not be prime.

• The remainder of the division of the square of a prime number p ≥ 5 divided by 24 is 1.

• For prime numbers p > 3, p2 - 1 is divisible by 24.

• If a and b are any two odd primes then a2 - b2 is composite. Also, a2 + b2 is composite.

• The remainder of the division of the square of a prime number p ≥ 5 divided by 12 is 1

Shortcut to Test for Prime numbers

• Step 1 : A number below 49 is prime if it is not divisible by 3 (Except numbers is even or ending with 0 / 5).

• Step 2 : A number between 49 - 121 is prime if it is not divisible by 3 (Except even numbers or ending with 0 / 5 or the numbers 77, 91, 119).

• Step 3 : A number between 121 - 169 is prime if it is not divisible by 3 (Except even numbers or numbers ending with 0 / 5 and 133, 143, 161).

Co prime or relatively prime: Two numbers are said to be relatively prime if they have only 1 as the common factor. E.g: (4,5); (8,9). If a number is divisible by p,q where both are co primes then it is also divisible by p*q;

Test for Prime numbers: To find out if a number 'N' greater than 100 is prime; we find out the number 'x' whose square is as close but greater than that number. If the prime numbers below 'x' are not dividing 'N' then it is prime.
E.g: 191 is prime since 14^2 = 196 and prime numbers below 14 are 2,3,5,7,11,13 don't divide 191.

Multiplication short cuts:
A) a * ( b + c ) = a*b + a*c
B) a * ( b - c) = a*b - a*c;

For shortcut to solving tough looking problems use above tricks:
345345 * 9999 = 345345 * (1000 - 1) which is simpler.

Multiplication of a number by 5^n: Add 'n' zeros to the right of the number and divide it by 2^n.
E.g: 975436 * 625 = 975436 * 5^4 = 9754360000/2^4

Ans .

3572021

1. Explanation :

Let x - 1936248=1635773. Then, x = 1635773 + 1936248 = 3572021.

Ans .

5526

1. Explanation :

Let 8597 - x = 7429 - 4358. Then, x = (8597 + 4358) - 7429 = 12955 - 7429 = 5526.

Ans .

9

1. Explanation :

When we add the numbers together, at the units place we have 9 + 7 + 8 = 24. We carried the 2 to the tens place and got P + Q + R + 2, As we need 11 at the next digit we need to carry forward 1. So P + Q + R + 2 = 11 and we get Q as 9. P = R = 0

Ans .

57928256595

1. Explanation :

5793405 x 9999 = 5793405(10000-1) = 57934050000-5793405 = 57928256595

Ans .

524673750

1. Explanation :

839478 x 625 = 839478 x 54 = 8394780000 / 16 = 524673750.

Ans .

986000

1. Explanation :

986 x 137 + 986 x 863 = 986 x (137 + 863) = 986 x 1000 = 986000

Ans .

98300

1. Explanation :

983 x 207 - 983 x 107 = 983 x (207 - 107) = 983 x 100 = 98300

Ans .

2576025

1. Explanation :

i) 1605 x 1605 = (1605) 2 = (1600 + 5) 2 = (1600) 2 + (5) 2 + 2 x 1600 x 5 = 2560000 + 25 + 16000 = 2576025.

Ans .

1954404

1. Explanation :

1398 x 1398 - (1398) 2 = (1400 - 2) 2 = (1400) 2 + (2) 2 - 2 x 1400 x 2 = 1960000 + 4 - 5600 = 1954404.

Ans .

180338

1. Explanation :

(a2 + b2) = 1/2 [(a + b)2 + (a - b)2] (313)2 + (287)2 = 1/2 [(313 + 287)2 + (313 - 287)2] = 1/2 [(600)2 + (26)2] = 1/2 (360000 + 676) = 180338.

Ans .

761200

1. Explanation :

= (896)2 - (204)2 = (896 + 204) (896 - 204) = 1100 x 692 = 761200.

Ans .

251001

1. Explanation :

(ii) Given exp = (387)2 + (114)2 + (2 x 387 x 114) = a2 + b2 + 2ab, where a = 387, b = 114 = (a+b)2 = (387 + 114)2 = (501)2 = 251001.

Ans .

169

1. Explanation :

Given exp = (81)2 + (68)2 – 2 x 81 x 68 = a2 + b2 – 2ab, Where a = 81, b = 68 = (a-b)2 = (81 – 68)2 = (13)2 = 169

Ans .

yes

1. Explanation :

Sum of digits in 541326 = (5 + 4 + 1 + 3 + 2 + 6) = 21, which is divisible by 3. Hence, 541326 is divisible by 3

Ans .

no

1. Explanation :

Sum of digits in 5967013 =(5+9 + 6 + 7 + 0+1 +3) = 31, which is not divisible by 3. Hence, 5967013 is not divisible by 3.

Ans .

2

1. Explanation :

Let the missing digit be x. Sum of digits = (1 + 9 + 7 + x + 5 + 4 + 6 + 2) = (34 + x). For (34 + x) to be divisible by 9, x must be replaced by 2 . Hence, the digit in place of * must be 2.

Ans .

no

1. Explanation :

The number formed by the last two digits in the given number is 94, which is not divisible by 4. Hence, 67920594 is not divisible by 4.

Ans .

yes

1. Explanation :

The number formed by the last two digits in the given number is 72, which is divisible by 4. Hence, 618703572 is divisible by 4.

Ans .

4 and 0

1. Explanation :

Since the given number is divisible by 5, so 0 or 5 must come in place of \$. But, a number ending with 5 is never divisible by 8. So, 0 will replace \$. Now, the number formed by the last three digits is 4*0, which becomes divisible by 8, if * is replaced by 4. Hence, digits in place of * and \$ are 4 and 0 respectively.

Ans .

yes

1. Explanation :

(Sum of digits at odd places) - (Sum of digits at even places) = (8 + 7 + 3 + 4) - (1 + 2 + 8) = 11, which is divisible by 11. Hence, 4832718 is divisible by 11.

Ans .

yes

1. Explanation :

24 = 3 x 8, where 3 and 8 are co-primes. The sum of the digits in the given number is 36, which is divisible by 3. So, the given number is divisible by 3. The number formed by the last 3 digits of the given number is 744, which is divisible by 8. So, the given number is divisible by 8. Thus, the given number is divisible by both 3 and 8, where 3 and 8 are co-primes. So, it is divisible by 3 x 8, i.e., 24.

Ans .

2

1. Explanation :

On dividing 3000 by 19, we get 17 as remainder. Number to be added = (19 - 17) = 2.

Ans .

3

1. Explanation :

On dividing 3105 by 21, we get 18 as remainder. So Number to be added to 3105 = (21 - 18) = 3. Hence, required number = 3105 + 3 = 3108

Ans .

9

1. Explanation :

On dividing the given number by 342, let k be the quotient and 47 as remainder. Then, number – 342k + 47 = (19 x 18k + 19 x 2 + 9) = 19 (18k + 2) + 9. The given number when divided by 19, gives (18k + 2) as quotient and 9 as remainder.

Ans .

100011

1. Explanation :

Smallest number of 6 digits is 100000. On dividing 100000 by 111, we get 100 as remainder. So Number to be added = (111 - 100) = 11. Hence, required number = 100011

Ans .

179

1. Explanation :

Divisor = (Dividend - Remainder ) / Quotient = ( 15968-37 ) / 89 = 179

Ans .

6, 4, 2

1. Explanation :

### Theorem of Divisibility

Calculate the number of factors of a number

• Step 1 : Calculate the factors of the number "X" less than or equal to its square root or if its not a perfect square than less than or equal to a number whose square is less but closest to it. Example: For 80 it will be 1 - 8 as 9 is 81, for 25 its 1 - 5. For 99 its 1 - 9.

• Step 2 : for factors above the square root limit, we just find the numbers with whom we can multiply the factors obtained from Step 1 and get the number "X". Example : For factors of 80, we find factors from 1 - 8, i.e. 1,2,4,5,8 then for factors above square root limit we get 80,40,20,16,10. As 1 * 80, 40 * 2, 20 * 4, 16 * 5 and 10 * 8.

Calculate the Sum of factors of a number:

Step 1:  Get prime factors of a number say 240

240 = 24 * 31 * 51

Step 2: Sum of factors formula is

240 = (20 + 21 + 22 + 23 + 24) * (30 + 31) * (50 + 51)

Step 3: 31*4*6 = 744

Calculate the Number of factors of a number:

Step 1: Get the prime factors of a number

240 = 24 * 31 * 51

Step 2: Number of factors of a number.

Number of factors = ( 4 + 1 ) * ( 1 + 1) * ( 1 + 1) = 5 * 2 * 2 = 20

Thus the powers of the numbers are increased by one and multiplied.

Calculate the sum and number of even factors of a number :

Numbers which do not have any power of 2 in their prime factor will have even factors equal to 0.

Step 1: Get the prime factors of a number

240 = 24 * 31 * 51

Step 2: Sum of even factors
Sum = (21 + 22 + 23 + 24) * (30 + 31) * (50 + 51)
Number of even factors = 4 * 2 * 2 = 16
Thus the powers of the numbers are increased by one and multiplied except 2.

Calculate the sum and number of odd factors of a number:

Step 1: Get the prime factors of a number

240 = 24 * 31 * 51

Step 2: Sum of odd factors
Sum = (20) * (30 + 31) * (50 + 51)
Number of odd factors = 1 * 2 * 2 = 4
Thus the powers of the numbers are increased by one and multiplied except 2.

Calculate the sum and number of factors of a number satisfying a condition:

Step 1:  Get prime factors of a number say 240

30 = 21 * 31 * 51

Step 2: Sum of factors formula is

30 = (20 + 21 ) * (30 + 31) * (50 + 51)

which is expanded as
= 20*30*50  + 20*30*50 + 20*31*51 + 20*31*51 + 21*30*50 + 21*30*50 + 21*31*51 + 21*31*51

These are all the factors of the number, We can apply any condition we want and remove unnecessary of them viz. remove factors that are perfect squares, not perfect squares, between higher and lower limit etc

Ans .

18

1. Explanation :

2450 = 50 * 49 = 21 * 52 * 72
Sum and number of all factors:
Sum of factors = (20 + 21) (50 + 51 + 52) (70 + 71 + 72) Number of factors = 2 * 3 * 3 = 18

Ans .

9

1. Explanation :

Sum of all even factors:
(21) (50 + 51 + 52) (70 + 71 + 72)
Number of even factors = 1 * 3 * 3 = 9

Ans .

9

1. Explanation :

Sum of all odd factors:
(20) (50 + 51 + 52) (70 + 71 + 72)
Number of odd factors = 1 * 3 * 3 = 9

Ans .

12

1. Explanation :

Sum of factors divisible by 5:
(20 + 21) (51 + 52) (70 + 71 + 72)
Number of factors divisible by 5 = 2 * 2 * 3 = 12

Ans .

8

1. Explanation :

Sum of factors divisible by 35:
(20 + 21) (51 + 52) (71 + 72)
Number of factors divisible by 5 = 2 * 2 * 2= 8

Ans .

4

1. Explanation :

Sum of factors divisible by 245:
(20 + 21) (51 + 52) (72)
Number of factors divisible by 5 = 2 * 2 * 1= 4

Ans .

54

1. Explanation :

Sum and number of all factors:
Sum of factors = (20 + 21+ 22 + 23 + 24 + 25) (30 + 31 + 32) (50 + 51 + 52)
Number of factors = 6 * 3 * 3 = 54

Ans .

30

1. Explanation :

Sum and number of all even factors:
Sum of factors = (21+ 22 + 23 + 24 + 25) (30 + 31 + 32) (50 + 51 + 52)
Number of factors = 5 * 3 * 3 = 30

Ans .

9

1. Explanation :

Sum and number of all odd factors:
Sum of factors = (20) (30 + 31 + 32) (50 + 51 + 52)
Number of factors = 1 * 3 * 3 = 9

Ans .

18

1. Explanation :

Sum and number of factors divisible by 25:
Sum of factors = (20 + 21+ 22 + 23 + 24 + 25) (30 + 31 + 32) (52)
Number of factors = 6 * 3 * 1 = 18

Ans .

18

1. Explanation :

Sum and number of all factors divisible by 40:
Sum of factors = (23 + 24 + 25) (30 + 31 + 32) (51 + 52)
Number of factors = 3 * 3 * 2 = 18

Ans .

10

1. Explanation :

Sum and number of factors divisible by 150:
Sum of factors = (21+ 22 + 23 + 24 + 25) ( 31 + 32) (52)
Number of factors = 5 * 2 * 1 = 10

Ans .

44

1. Explanation :

Sum and number of factors NOT divisible by 150 = Total factors - Factors divisible by 150 = 54 - 10 = 44

Ans .

12

1. Explanation :

Sum and number of all factors who are perfect squares:
Sum of factors = (20 + 22 + 24 ) (30 + 32) (50 + 52)
Number of factors = 3 * 2 * 2 = 12

New Age Consultants have three consultants Gyani, Medha and Buddhi. The sum of the number of projects handled by Gyani and Buddhi individually is equal to the number of projects in which Medha is involved. All three consultants are involved together in 6 projects. Gyani works with Medha in 14 projects. Buddhi has 2 projects with Medha but without Gyani, and 3 projects with Gyani but without Medha. The total number of projects for New Age Consultants is one less than twice the number of projects in which more than one consultant is involved.

Q. What is the number of projects in which Gyani alone is involved?

1. Uniquely equal to zero.

2. Uniquely equal to 1.

3. Uniquely equal to 4

4. Cannot be determined uniquely.

Ans . D

1. Putting the value of M in either equation, we get G + B = 17

2. Hence neither of two can be uniquely determined.

Q. What is the number of projects in which Medha alone is involved?

1. Uniquely equal to zero.

2. Uniquely equal to 1.

3. Uniquely equal to 4

4. Cannot be determined uniquely.

Ans . B

1. G + B = M + 16 Also, M + B + G + 19 = (2 * 19) – 1

2. i.e. (G + B) = 18 – M Thus, M + 16 = 18 – M

3. i.e. M = 1

### Quiz

Score more than 80% marks and move ahead else stay back and read again!