Important Terms:
Addition =
Minimum/Smallest =
Series =
Subtraction =
Divisor =
Consecutive =
Multiplication =
Dividend =
Product =
Division =
Quotient =
Rational Number =
Natural Number =
Remainder =
Irrational Number =
Integer =
Multiple =
Perfect Square Number =
Whole Number =
Factor =
Perfect Cube Number =
Decimal = দ
Divisible =
Positive =
Even Number =
Face Value =
Negative =
Odd Number =
Local Value/Place Value =
Least Value =
Prime Number =
Approximate Value =
Non- negative =
Greatest/Highest =
Sequence =
Fraction =
/
দ
দ
 Natural Number: Counting numbers 1 , 2 , 3, 4, 5, …. are known as natural number. The set of all natural
number can be represented by N = {1, 2, 3, 4, 5……..}
 Whole Numbers: If we include 0 among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5,….are called whole
numbers. The set of whole numbers can be represented by W = {0, 1, 2, 3, 4, 5…..}
Clearly, every natural number is a whole number but 0 is a whole number which is not a natural number.
 Integers: All counting numbers and their negatives including zero are known as integers. The set of integers can
be represented by Z or I = {…, - 4, -3, -2, - 1, 0, 1, 2, 3, 4, …}
 Positive Integers: The set I+ = {1, 2, 3, 4, …} is the set of all positive integers. Clearly, positive Integers and
natural numbers are synonyms.
 Negative Integers: The set I- = {- 1, -2, -3,…} is the set of all negative integers. 0 is neither positive nor negative.
 None-negative Integers: The set {0, 1, 2, 3,…} is the set of all non-negative integers.
 Rational Numbers: The numbers of the form
number, e.g. ,
,
That is Q = {x:x =
,-
, , - , etc. The set of all rational numbers is denoted by Q.
; p, q
I, q ≠ 0}
Since every natural number ‘a’ can be written as
written as
, where p and q are integers and q ≠ 0, are known as rational
, every natural number is a rational number. Since 0 can be
and every non-zero integer ‘a’ can be written as , every integer is a rational number. Every rational
number has a particular characteristic that when expressed in decimal form is expressible either in terminating
decimals or in non-terminating repeating decimals.
For example,
= 0.2,
= 0.333….,
= 3.1428714287,
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= 0.181818….., etc.
Page 2
The recurring decimals have been given a short notation as –
0.333…… = 0. ̅
4.1555…… = 4.0 ̅
0.323232...… = 0. ̅̅̅̅
 Irrational Numbers: Those numbers which when expressed in decimal form are neither terminating nor
repeating decimal are known as irrational number. e.g. ,
, etc.
Note that the exact value of
is not
.
is rational while
is irrational number.
is approximate value of .
Similarly, 3.14 is not an exact value of it.
 Real Number: The rational and irrational numbers combined together to form real numbers, e.g.,
4+
,
, etc. are real numbers. The set of all real number is denoted by R.
Note that the sum, difference or product a rational and irrational number is irrational, e.g. 3+
7
, ,- ,
, 4-
, ,4
,-
are all irrational.
 Factor: A factor is one of two or more numbers that divides a given number without a remainder.
For example, 4 is a factor of 12.
 Multiple: A multiple is a number that can be divided by another number a certain number of times without a
remainder.
For example, 25 is a multiple of 5.
 Prime Numbers: A natural number other than 1, is a prime number if it is divisible by 1 and itself only.
For example, each of the numbers 2, 3, 5, 7, etc. are prime numbers.
 Composite Numbers: Natural number grater than 1 which are not prime, are known as composite numbers.
For example, each of the numbers 4, 6, 8, 9, 12, etc. are composite numbers.
1. The number 1 is neither a prime number nor a composite number.
2. 2 is the only even number which is prime.
3. Prime number up to 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79, 83, 89, 97, i.e. 25 prime numbers between 1 and 100.
4. Two numbers which have only 1 as the common factor are called co-primes or
relatively prime to each other, e.g., 3 and 5 are co-primes.

Note that the numbers which are relatively prime need not necessarily be
prime numbers, e.g., 16 and 17 are relatively prime, although 16 is not a
prime number.
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 Square: A number multiplied by itself is known as the square of a given number.
For example, square of 6 is 6 × 6 = 36.
 Perfect Square Number: A perfect square is a number that can be expressed as the product of an integer by itself
or as the second exponent of an integer. For example, 25 is a perfect square because it is the product of integer 5
by itself, 5 × 5 = 25.
 Square Root: Square root of a given number is that number, which when multiplied by itself is equal to the given
number.
For example, square root of 81 is 9, because
= 9 × 9 = 81.
The square root of a number is denoted by the symbol , called radical sign.
Thus
= 9,
= 8 and, so on.
Note,
=1
 Method of Finding a Square Root:
 Prime Factorization Method:
Step Ⅰ: Find the prime factors of a given number.
Step Ⅱ: Group the factors in pairs.
Step Ⅲ: Take one number form of each pair of factors. Multiply them together.
The product thus derived the square root of the given number.
For example, 4761 = (23 × 23) × (3 × 3)
∴
= 23 × 3 = 69
# N:B: The above method is used when a given number is a perfect square or when every prime factor of that
number is repeated twice.
 Method of Division: This method is used when the number is large and the factors cannot be easily determined.
Step Ⅰ: The digits of a number, whose square root is required are separated into periods of two beginning from the
right. The last period may be either or a pair.
Step Ⅱ: Find a number (here, 4) whose square may be equal to or less than the first period (here, 22).
Step Ⅲ: Find out the reminder (here, 6) and bring down the next period (here, 65)
Step Ⅳ: Double the quotient (here, 4) and write to the left (here, 8)
Step Ⅴ: The divisor of this stage will be equal to the above sum (here, 8) with the quotient of this stage (here, 7)
suffixed to it (here, 87).
Step ⅤⅠ: Repeat this process (step Ⅳ and step Ⅴ) till all the periods get exhausted. The quotient (here, 476) is
equal the square root of the given number (here, 226576).
For example,
̅̅̅̅ ̅̅̅̅ ̅̅̅̅
16
87
476
665
604
946
5676
5676
0
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 Cube: Cube of a number is obtained by multiplying the number itself thrice.
For example, 27 is the cube of 3 as 27 = 3 × 3 × 3.
 Cube Root: The cube root of a given number is that number, which, when raised to the third power, produces the
given number, that is, the cube root of a number x is the number whose cube is x. The cube root of x is written as
For example, cube root of 64 is 4 as 4 × 4 × 4 = 64.
 Methods to Find Cube Root:
 Method of Factorization:
Step Ⅰ: Write the given number as product of prime factors.
Step Ⅱ: Take the product of prime numbers, choosing one out of three of each type.
This product gives the cube root of the given number.
For example, 42875 = (5 × 5 × 5) × (7 × 7 × 7)
∴
= 5 × 7 = 35
 Short-cut Method to Find Cube Roots of Exact Cubes Consisting of up to 6 Digits:
Before we discussed the method to find the cube roots of exact cubes, the following to remarks are useful and
must be keep in mind.
3
3
3
3
3
3
3
3
1.
1 = 1; 2 = 8; 3 = 27; 4 = 64; 5 = 125; 6 = 216; 7 = 343; 8 = 512;
3
3
9 = 729; 10 = 1000
2.
If the cubes ends in 1, then the cube root ends in 1.
If the cubes ends in 2, then the cube root ends in 8.
If the cubes ends in 3, then the cube root ends in 7.
If the cubes ends in 4, then the cube root ends in 4.
If the cubes ends in 5, then the cube root ends in 5.
If the cubes ends in 6, then the cube root ends in 6.
If the cubes ends in 7, then the cube root ends in 3.
If the cubes ends in 8, then the cube root ends in 2.
If the cubes ends in 9, then the cube root ends in 9.
If the cubes ends in 0, then the cube root ends in 0.
Clearly, from the given, 1
1, 2
8, 3
7, 4
4, 5
5, 6
6, 7
3, 8
2, 9
9, 0
0
The method of finding the cube root of a number up to 6 digits, which is actually a cube of some number
consisting of 2 digits, is best illustrated with the help of the following examples.
 Even Number: All those numbers which are exactly divisible by 2 are called even numbers, e.g., 2, 6, 8, 10, etc.
are even numbers.
 Odd Number: All those numbers which are not exactly divisible by 2 are called odd numbers, e.g., 1, 3, 5, 7, etc.
are odd numbers.
Let us assume N as an integer. If there exists an integer P such that N = 2P + 1, then N is an odd number. If there
exists an integer P such that N = 2P, then N is an even number.
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Table of Odd and Even
Even + Even = 2 × Even = multiple of 2 = Even
Even + Odd = Odd
Odd + Odd = 2 × Odd = multiple of 2 = Even
2
Even × Even = Even = Even
Even
= Even
Even × Odd = 2 × Odd = multiple of 2 = Even
2
Even
Odd × Odd = Odd = Odd
= Odd
3
Odd
Odd × Odd × Odd = Odd = Odd
= Odd
 Tests of divisibility:
1. Divisibility by 2: A number is divisible by 2 if the unit’s digit is zero or divisible by 2.
For example, 4, 12, 30, 18, 102, etc., are all divisible by 2.
2. Divisibility by 3: A number is divisible by 3 if the sum of digits in the number is divisible by 3.
For example, the number 3792 is divisible by 3 since 3 + 7 + 9 + 2 = 21, which is divisible by 3.
3. Divisibility by 4: A number is divisible by 4 if the number formed by the last two digits (ten’s digit and unit’s
digit) is divisible by 4 or are both zero.
For example, the number 2616 is divisible by 4 since 16 is divisible by 4.
4. Divisibility by 5: A number is divisible by 5 if the unit’s digit in the number is 0 or 5.
For example, 13520, 7805, 640, 745, etc., are all divisible by 5.
5. Divisibility by 6: A number is divisible by 6 if the number is even and sum of its digits is divisible by 3.
For example, the number 4518 is divisible by 6 since it is even and sum of its digits 4 + 5 + 1 + 8 = 18 is divisible
by 3.
6. Divisibility by 7: The unit digit of the given number is doubled and then it is subtracted from the number
obtained after omitting the unit digit. If the remainder is divisible by 7, then the given number is also divisible by
7.
For example, consider the number 448. On doubling the unit digit 8 of 448 we get 16. Then, 44 – 16 = 28. Since
28 is divisible by 7, 448 is divisible by 7.
7. Divisibility by 8: A number is divisible by 8, if the number formed by the last 3 digits is divisible by 8.
For example, the number 41784 is divisible by 8 as the number formed by last three digits, i.e., 784 is divisible by
8.
8. Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
For example, the number 19044 is divisible by 9 as the sum of its digits 1 + 9 + 0 + 4 + 4 = 18 is divisible by 9.
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9. Divisibility by 10: A number is divisible by 10, if it ends in zero.
For example, the last digit of 580 is zero, therefore, 580 is divisible by 10.
10. Divisibility by 11: A number is divisible by 11 if the difference of the sum of the digits at odd places and sum
of the digits at even places is either zero or divisible by 11.
For example, in the number 38797, the sum of the digits at odd places is 3 + 7 + 7 = 17 and the sum of the digits at
even places is 8 + 9 = 17. The difference is 17 – 17 = 0, so the number is divisible by 11.
11. Divisibility by 12: A number is divisible by 12 if it is divisible by 3 and 4.
12. Divisibility by 13: (A + 4B), where B is the unit’s place digit and A is all the remaining digits.
For example, let us check the divisibility of 1404 by 13. Here, A = 140 and B = 4, then A + 4B = 140 + 4 × 4 =
156. This 156 divisible by 13 and therefore 1404 will be divisible by 13.
13. Divisibility by 17: (A - 5B), where B is the unit’s place digit and A is all the remaining digits.
For example, let us check the divisibility of 1632 by 17. Here, A = 163 and B = 2, then A - 5B = 163 - 2 × 5 =
153. This 153 divisible by 17 and therefore 1632 will be divisible by 17.
14. Divisibility by 18: An even number satisfying the divisibility test of 9 is divisible by 18.
15. Divisibility by 19: (A + 2B), where B is the unit’s place digit and A is all the remaining digits.
For example, let us check the divisibility of 1634 by 19. Here, A = 163 and B = 4, then A + 2B = 163 + 2 × 4 =
171. This 171 divisible by 19 and therefore 1634 will be divisible by 19.
16. Divisibility by 25: A number is divisible by 25 if the number formed by the last two digits is divisible by 25
or the last two digits are zero.
For example, the number 13675 is divisible by 25 as the number formed by the last two digits is 75 which is
divisible by 25.
17. Divisibility by 88: A number is divisible by 88 if it is divisible by 11 and 8.
18. Divisibility by 125: A number is divisible by 125 if the number formed by the last three digits is divisible by
125 or the last three digits are zero.
For example, the number 5250 is divisible by 125 as 250 is divisible by 125.
 Dividend = Quotient × Divisor + Remainder:
The basic framework of remainder are as follows:
a. If N is a number divisible by 7, it can be written as 7K = N, where K is the quotient.
b. When N is divided by 7, remainder obtained is 3. Therefore, it can be written as 7K + 3 = N, where K is the
quotient.
c. When N is divided by 7, remainder obtained is 3 and it is equivalent of saying remainder obtained is (-4) when
divided by 7, Remainder obtained is 3 = N is 3 more than a multiple of 7, Therefore, N is 4 shorts of another
multiple of 7. Therefore, remainder obtained = - 4.
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d. When N is divided by 8, different remainders can be obtained. They are 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4 (5 different
remainders) are obtained.
 Two Digits Number: (A + 10B) is two digits number, where A is in the place in unit digit and B is in the place
in tens digit and the digit interchanging number is (B + 10A).
 Concepts of Digits:
986 7
এ
(Units)
দ
(tens)
(hundreds)
সহস্র (Thousands)
 9 8 6 7. 2 3 1 5
ten thousandths (দ সহস্ররাংশ)
thousandths(সহস্ররাংশ)
hundredths (শতরাংশ)
tenths (দশমরাংশ)
Class Work
1. How many ‘8’ will you pass on the way when you count from 1 to 100?
a. 10
b. 11
c. 20
d. 70
(Bangladesh Bank: AD– 2013)
e. 80
d. n² - 1
(UCBL - 2017: Exim Bank - 2020)
e. 2(n + 3)
2. If n is even, which of the following cannot be odd?
a. n + 3
b. 3(n + 1)
c. 2(n + 3) + 3
3. If a, b, and c are consecutive positive integers and a < b < c, which of the following must be an odd integer?
(IBA - BBA: 07;Trust Bank : MTO - 2015)
a. abc
b. a + b + c
c. a + bc
d. a(b + c)
e. (a + b)(b + c)
4. x and y are integers and x = 32y + 15, which of the following must be an odd integer? (FSIB: PO - 2014)
a. xy
b. x + y
c. x + 2y
d. None of these
5. Which one of the following numbers is exactly divisible by 3?
a. 2135
b. 2033
c. 2155
d. 8124
6. Which of the following number is divisible by 24?
a. 35718
b. 63810
c. 537804
d. 3125736
7. The number 87715938* is divisible by 4. The unknown non - zero digit marked as * will be –
(BD house building - 2017)
a. 4
b. 3
c. 2
d. 6
8. Which one of the following numbers is exactly divisible by 11?
(Pubali Bank Lts. Trainee Assistant Teller 2017)
a. 235641
b. 245642
c. 315624
d. 415624
9. If n is an integer greater than 5, which of the following must be divisible by 4? (IBA - MBA : 55th intake)
a. n(n + 2)(n -2)
b. (n + 4)(n - 2)
c. n(2n + 4)(n - 1)
d. (n + 5)(n -2)
e. None of these
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10. who many 3- digit numbers exactly divisible by 6?
a. 140
b. 150
c. 160
d. 170
11. How many integers from 1 to 1000 are divisible by 30 but not by 16?
[Officer (BB)-`18, `19; Rupali- `19;Sr. Officer (Sonali)-`18; Officer (Cash)- Agrani-`17]
a. 29
b. 31
c. 32
d. 38
12. Which of the following is perfect square number? (IBA - MBA: 54th intake)
a. 314723
b. 204314
c. 322624
d. 5382532
e. 363272
13. What is the square root of 114921? (IBA - MBA: 55th intake)
a. 324
b. 333
c. 336
e. None of these
d. 339
14. What is the least number by which 2450 must be multiplied in order to get a perfect square?
(Southeast Bank: PO – 2017)
a. 2
b. 3
c. 4
d. 8
15. What is the smallest number to be subtracted from 549162 in order to make it a perfect square?
(Janata Bank - 2015)
a. 28
b. 36
c. 62
d. 81
16. Which of the following numbers is a prime? (Combined bank: SO - 2000)
a. 49
b. 51
c. 53
d. 55
17. Which of the following is a prime number?
a. 161
b. 221
c. 373
d. 437
18. Find the number of divisors of 72. (26th BCS)
a. 9
b. 10
c. 11
d. 12
19. How many odd divisors does 540 have? (IBBL (PO) – 2017)
a. 8
b. 12
c. 16
d. 24
20. Which of the following numbers has odd numbers of divisors? (RBL- 2013; BCS -2016)
a.2048
b. 1024
c. 512
d. 48
21. Find the value of 0!
a. 0
b. 1
c. Infinity
22. What is the greatest power of 5 which can divide 80! exactly?
a. 13
b. 15
c. 16
d. undefined
e. None of these
d. 19
e. None of these
23. Suppose A is the product of all integers from 2 to 20 inclusive. If 2ⁿ is a factor of A, what is the greatest
possible value for integer n? (DBBL - 2019)
a. 10
b. 16
c. 18
d. 20
24. If (125)14
a. 24
(48)8 = x; Then how many trailing zeros does ‘x’ have? (IBA -MBA: 63th intake)
b. 32
c. 42
d. 54
e. None of these
25. The quantity 2⁶5⁷3⁷ will end how many zeros? (IBA - BBA: 2014 - 15)
a. 4
b. 5
c. 6
d. 7
e. None of these
26. What is the unit digits in the product (648 759 413 676)? (Janata Bank: EO 2017)
a. 4
b. 12
c. 8
d. 6
27. If n = 7⁹ - 6, what is the units of n? (IFIC Bank - 2013, IBA)
a. 7
b. 2
c. 0
d. 1
e. None of these
28. What is the unit digit of 4218? (IBA – BBA: 2004-05)
a. 2
b. 4
c. 6
d. 8
e. None of these
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29. If n = (33)43 + (43)33, what is the unit digit of n?
a. 0
b. 3
c. 5
d. 7
e. None of these
30. If x= 106 109 111, Then what is the remainder when x is divided by 15? (IBA -MBA: 63th intake)
a. 1
b. 3
c. 9
d. 11
e. None of these
31. What is the remainder when 82000 is divided by 7? (Shahjalal Islami Bank - 2013)
a. 1
b. 2
c. 3
d. 4
e. 6
32. What is the remainder when 3²⁴ is divided by 5? (Bank Asia - MTO: 2017; Madhumati Bank - PO:2018)
a. 0
b. 1
c. 2
d. 3
e. None of these
33. When positive integer x is divided by positive integer y, the remainder is 9. If = 96.12, what is the value of
y? (IBA - BBA: 2008 - 09 ; GMAT O GUIDE - 154)
a. 96
b. 75
c. 48
d. 25
e. 12
34. A number when divided successively by 4 and 5 leaves remainders 1 and 4 respectively. When it is
successively divided 5 and 4, what will be the respectively remainders? (Janata Bank – 2017)
a. 2, 4
b. 2, 3
c. 3, 4
d. 3, 2
e. None of these
35. In dividing a number by 585, a student employed the method of short division. He divided the number
successively by 5, 9 and 13 (factors of 585) and got the remainders 4, 8, 12 respectively. If he had divided the
number by 585, what would have been the remainder? (IBA-BBA: 2013 - 14)
a. 24
b. 1444
c. 292
d. 584
e. None of these
36. A two digit number is such that the product of the digits is 8. When 18 is added to the number, then the
digits are reversed. The number is - (Jamuna Bank: MTO - 2013)
a. 18
b. 24
c. 42
d. 81
37. The difference between a two digit number & the number obtained by interchanging the positions of its
digits is 36. What is the difference between the two digits of that number? (BB-AD: 2012, Exim Bank-MTO:
2013, Premier Bank-TJO: 2020)
a. 3
b. 4
c. 6
d. 9
38. The sum of the digits of two-digit number is 10, while when the digits are reversed, the number decrease by
54. Find the changed number. (Combined 5 Bank– 2019)
a. 19
b. 46
c. 37
d. 28
e. None of these
39. The difference of two number is 1365. On dividing the larger number by the smaller, we get 6 as quotient
and 15 as remainder. What is the smaller number? (Basic Bank Ltd. Asst. Manager 2018)
a. 270
b. 1270
c. 350
d. 720
40. If A = 8B + 22, and B is a positive integer, then A is not divisible by which of the following? (BKB-SO: 2015)
a. 2
b. 4
c. 6
d. 7
e. None of these
Questions for Practice
41. If y = 4¹⁰ + 4¹¹ + 4¹² + 4¹³, then y is divisible by which number? (IBA - MBA: 55th intake)
a. 12
b. 13
c. 17
d. 19
e. None of these
42. What is the greatest positive integer n such that 2ⁿ is a factor of 12¹⁰? (IBA - BBA : 2001 - 02)
a. 10
b. 12
c. 16
d. 20
e. 60
43. Find number of zeros at the end in 18! + 19!
a. 3
b. 4
c. 5
d. 6
e. 2
44. If x is a product of 4 consecutive integers, and x is divisible by 11, which of the following is not necessarily a
divisor of x? (IBA-MBA: 59th intake)
a. 12
b. 22
c. 24
d. 33
e. None of these
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45. For an integer n, n! = n(n – 1)(n – 2)………3.2.1, then what is the unit digit of 1! + 2! + 3! + 4! +…….+ 101!?
a. 1
b. 3
c. 5
d. 6
e. None of these
46. If An= 7ⁿ - 1, what is the unit digit of A₃₃?
a. 2
b. 3
c. 5
d. 6
e. 9
47. The difference between a two-digit number and the number obtained by interchanging the digits is 36.
What is the difference between the sum and the difference of the digits of the number if the ratio between the
digits of the number is 1 : 2?(DBBL. : PO – `12)
a. 9
b. 6
c. 7
d. 8
e. None of these
48. If k is a multiple of 24 but not a multiple of 16, which of the following cannot be an integer?
a.
b.
c.
d.
e.
49. When x is divided by 10, the quotient is y with a remainder of 4. If x and y are both positive integers, what
is the remainder when x is divided by 5?
a. 0
b. 1
c. 2
d. 3
e. 4
50. How many prime numbers between 1 and 100 factors of 7,150? (O - GMAT - 205)
a. One
b. Two
c. Three
d. Four
e. Five
51. The least number of 4 digits which is a perfect square, is: (Rupali Bank – 2013)
a. 1000
b. 1016
c. 1024
d. 1036
52. If *x is defined as the square of one - half of x, what is the value of
a.
b.
c.
?
d.
e. None of these
53. If 3 < x < 100, for how many values of x is the square of a prime number? (O - GMAT - 181)
a. Two
b. Three
c. Four
d. Five
e. Nine
54. If x, y, w and z correspond to four numbers - 3, , - 4 and 2 but not necessarily in the same order, what is the
largest possible value of the expression ( ) z2? (IBA – BBA: 2003-04)
a. 92
b. 36
c. 24
d. 12
e. None of these
55. In first 1000 natural numbers, who many integers exist such that they leave a remainder 4 when divided by
7 and a reminder 9 when divided by 11? (BKB 2017; BHBFC S O 2017)
a. 11
b. 13
c. 15
d. 17
Answer Sheet:
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(c) 17
46
(d) 6
51
(c) 1024
42
47
(d) 20
(d) 8
43
48
52
(c)
53
(b) 4
(c)
(b) Three
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44
49
(e) None of these
(e) 4
45
50
(b) 3
(c) Three
54
(e) None of these
55
( b ) 13
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