Hello Aspirants,
I am Rahul Sir, and over the years I have guided hundreds of banking aspirants preparing for IBPS PO, IBPS Clerk, SBI PO, SBI Clerk, RRB, and other competitive examinations. One topic that consistently appears in aptitude sections is Problems on Numbers. Many students find number-based questions tricky because they involve concepts from divisibility, remainders, factors, multiples, digit properties, and logical number patterns.
The good news is that these questions become much easier when you understand a few fundamental rules and shortcut techniques. In IBPS examinations, number problems are generally designed to test logical thinking rather than lengthy calculations. Therefore, students who know the right tricks can solve questions quickly and accurately.
In this article, I will explain the most important concepts, shortcuts, and exam-oriented strategies for solving Problems on Numbers. Each section includes practical examples similar to those asked in banking examinations. If you master these techniques and practice regularly, you can significantly improve your speed and accuracy in the quantitative aptitude section.
Let us begin with the essential concepts that every banking aspirant must know.
1. Understanding Number Systems
Problems on Numbers begin with understanding the number system. Every question is built around basic properties of numbers.
The main categories include:
- Natural Numbers: 1, 2, 3, 4…
- Whole Numbers: 0, 1, 2, 3…
- Integers: Negative and positive numbers including zero
- Even Numbers: Divisible by 2
- Odd Numbers: Not divisible by 2
- Prime Numbers: Numbers having exactly two factors
- Composite Numbers: Numbers having more than two factors
Understanding these categories helps identify patterns quickly.
Example
Which of the following is a prime number?
A) 39
B) 51
C) 53
D) 57
Check divisibility:
53 is not divisible by 2, 3, 5, or 7.
Therefore, 53 is prime.
IBPS Tip
Memorize prime numbers up to 100. Many number-based questions become easier when you instantly recognize prime numbers.
Shortcut
A number greater than 3 is prime only if it is not divisible by any prime number less than its square root.
Strong understanding of number classifications creates the foundation for advanced number problems in banking examinations.
2. Divisibility Rules You Must Know
Divisibility rules save valuable exam time and reduce calculations.
Important rules:
- Divisible by 2 → Last digit even
- Divisible by 3 → Sum of digits divisible by 3
- Divisible by 4 → Last two digits divisible by 4
- Divisible by 5 → Ends in 0 or 5
- Divisible by 6 → Divisible by 2 and 3
- Divisible by 8 → Last three digits divisible by 8
- Divisible by 9 → Sum of digits divisible by 9
- Divisible by 11 → Difference between alternate digit sums is 0 or multiple of 11
Example
Is 35784 divisible by 9?
3 + 5 + 7 + 8 + 4 = 27
27 is divisible by 9.
Therefore, 35784 is divisible by 9.
Example
Is 6524 divisible by 4?
Last two digits = 24
24 is divisible by 4.
Hence 6524 is divisible by 4.
IBPS Tip
Most remainder and factor questions can be solved quickly using divisibility rules without actual division.
Practice applying these rules mentally to improve speed.
3. Factors and Multiples Techniques
Factors and multiples form the basis of many aptitude questions.
Important Formula
If
N = pᵃ × qᵇ × rᶜ
Number of factors = (a+1)(b+1)(c+1)
Example
Find total factors of 72.
72 = 2³ × 3²
Factors = (3+1)(2+1)
= 4 × 3
= 12 factors
Multiple Concept
A multiple is obtained by multiplying a number by an integer.
Multiples of 12:
12, 24, 36, 48, 60…
Example
Find the least number divisible by both 12 and 18.
LCM(12,18)
12 = 2² × 3
18 = 2 × 3²
LCM = 2² × 3²
= 36
IBPS Tip
Questions involving common divisibility often require LCM and HCF concepts. Learning prime factorization speeds up calculations significantly.
4. Remainder Theorem Tricks
Remainder questions are extremely common in IBPS examinations.
Basic Formula:
Dividend = Divisor × Quotient + Remainder
Example
What is the remainder when 257 is divided by 6?
6 × 42 = 252
257 − 252 = 5
Remainder = 5
Shortcut Example
Find remainder when 2⁵ divided by 3.
2⁵ = 32
32 ÷ 3
Remainder = 2
Pattern Method
Find remainder when 7⁵ divided by 5.
Units digit pattern:
7¹ = 7 → 2 remainder
7² = 49 → 4 remainder
7³ → 3 remainder
7⁴ → 1 remainder
Cycle repeats.
7⁵ gives remainder 2.
IBPS Tip
Many large power questions can be solved using remainder cycles instead of calculating actual values.
This technique saves considerable time during exams.
5. Finding Unit Digits Quickly
Unit digit questions are favorites in banking exams.
Example
Find unit digit of 8⁵³.
Observe pattern:
8¹ = 8
8² = 64 → 4
8³ → 2
8⁴ → 6
Pattern repeats every 4 terms.
53 ÷ 4 gives remainder 1.
Therefore unit digit = 8.
Example
Unit digit of 3⁴²
Pattern:
3, 9, 7, 1
42 ÷ 4 gives remainder 2
Second digit in pattern = 9
Answer = 9
IBPS Tip
Memorize unit digit cycles for:
2, 3, 4, 7, 8, 9
These appear frequently in competitive examinations.
Mastering this concept allows solving questions within seconds.
6. HCF and LCM Shortcuts
HCF and LCM are essential topics in number problems.
Relationship Formula
HCF × LCM = Product of Numbers
Example
Two numbers are 12 and 18.
HCF = 6
LCM = 36
6 × 36 = 216
12 × 18 = 216
Verified.
Example
HCF of 48 and 72
48 = 2⁴ × 3
72 = 2³ × 3²
Common factors:
2³ × 3
= 24
IBPS Tip
Prime factorization is usually faster than repeated division.
Questions involving bells, clocks, intervals, or repeated events often use LCM concepts.
Regular practice makes these calculations effortless.
7. Digit-Based Number Problems
Many IBPS questions involve digits and their relationships.
Example
A two-digit number has digits whose sum is 9 and difference is 3. Find the number.
Let digits be x and y.
x + y = 9
x − y = 3
Adding:
2x = 12
x = 6
y = 3
Number = 63
Example
If digits of a two-digit number are reversed, the number decreases by 27. Difference between digits?
Difference = 27 ÷ 9
= 3
Shortcut
Reversing a two-digit number changes value by:
9 × Difference of digits
IBPS Tip
Memorizing such relationships saves time during algebraic number questions.
8. Consecutive Number Problems
Questions involving consecutive numbers are common.
Formulas
Sum of first n natural numbers:
n(n+1)/2
Example
Sum of first 20 natural numbers
20 × 21 / 2
= 210
Example
Three consecutive numbers have sum 72.
Let numbers be:
x−1, x, x+1
3x = 72
x = 24
Numbers = 23, 24, 25
IBPS Tip
Represent consecutive numbers using variables to simplify equations.
This approach reduces complexity and improves accuracy.
9. Number Series and Pattern Recognition
Pattern recognition helps solve advanced number questions.
Example
2, 6, 12, 20, 30, ?
Differences:
4, 6, 8, 10
Next difference = 12
30 + 12 = 42
Answer = 42
Example
1, 4, 9, 16, 25, ?
These are squares:
1², 2², 3², 4², 5²
Next = 6² = 36
IBPS Tip
Always check:
- Addition/Subtraction pattern
- Multiplication pattern
- Squares
- Cubes
- Prime numbers
Pattern recognition becomes faster with regular practice.
10. Exam Strategy for Problems on Numbers
Knowledge alone is not enough. Strategy is equally important.
Step 1
Read the question carefully.
Step 2
Identify the concept:
- Divisibility
- Factors
- Remainders
- Unit digits
- HCF/LCM
- Digit relations
Step 3
Apply shortcuts before attempting lengthy calculations.
Step 4
Use option elimination whenever possible.
Step 5
Avoid spending more than one minute on a single question.
Example
If a question asks whether a number is divisible by 9, do not divide. Use the digit sum rule.
This saves valuable exam time.
IBPS Success Formula
Concept Clarity + Shortcuts + Practice + Time Management
Students following this formula consistently perform better in quantitative aptitude sections.
Regular mock tests and sectional practice are essential for mastery.
How Teachers from OdTutor Can Help
At OdTutor, our experienced banking aptitude faculty help students master Problems on Numbers through concept-focused learning and exam-oriented practice. Our teachers explain divisibility rules, HCF-LCM shortcuts, remainder techniques, unit digit patterns, and advanced number concepts using easy-to-understand methods. Students receive live interactive classes, doubt-solving sessions, chapter-wise assignments, mock tests, and previous-year IBPS question practice. We focus on improving both speed and accuracy, which are crucial for IBPS PO and Clerk examinations. With personalized guidance, performance tracking, and structured preparation plans, OdTutor helps aspirants build confidence and maximize their scores in quantitative aptitude sections.