CAT 2025 :Number System Guide – Tricks & Examples
- RAHUL SINGH
- 2 days ago
- 5 min read
1. CAT 2025 and Number System
The Number System forms the cornerstone of all mathematical understanding and is a key area in many competitive exams such as CAT 2025 , GMAT, SSC, Bank PO, GRE, and other aptitude-based entrance tests. It not only improves one's problem-solving ability but also enhances speed and accuracy.
Many students fear the Number System because of its diversity and the need for strong foundational understanding. This guide aims to demystify the topic and provide readers with in-depth knowledge, shortcut tricks, solved examples, and practice exercises. We will cover classifications, arithmetic properties, divisibility, remainders, special numbers, and more in a step-by-step manner.
2. Classification of Numbers
2.1 Natural Numbers (N)
These are counting numbers: 1, 2, 3, 4...
Denoted by N
No inclusion of 0 or negative numbers
Used in everyday counting
2.2 Whole Numbers (W)
Includes natural numbers and 0
Set: 0, 1, 2, 3, 4...
Denoted by W
Important in set theory and basic calculations
2.3 Integers (Z)
Includes negative numbers, 0, and positive whole numbers
..., -3, -2, -1, 0, 1, 2, 3...
Denoted by Z (from German "Zahlen")
2.4 Rational Numbers (Q)
Numbers that can be expressed as p/q where q ≠ 0
Includes fractions, terminating decimals, and repeating decimals
Examples: 1/2, -3/4, 0.25, 2, 5.333...
2.5 Irrational Numbers
Cannot be expressed as a ratio of two integers
Decimal expansion is non-terminating and non-repeating
Examples: √2, √3, π, e
2.6 Real Numbers
All rational and irrational numbers together form the real number system
Represented on the number line
2.7 Imaginary and Complex Numbers
Imaginary Numbers: Involve √(-1) denoted as i
Complex Numbers: Of the form a + bi where a and b are real numbers
Example: 3 + 2i
3. Important Properties & Rules
3.1 Commutative Property
Applies to addition and multiplication
a + b = b + a and a × b = b × a
3.2 Associative Property
(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c)
3.3 Distributive Property
a × (b + c) = a×b + a×c
3.4 Identity Elements
Additive identity = 0
Multiplicative identity = 1
3.5 Inverse Elements
Additive inverse: a + (-a) = 0
Multiplicative inverse: a × (1/a) = 1 for a ≠ 0
4. Divisibility Rules (1 to 20)
Mastering divisibility rules saves valuable seconds in exams:
2: Last digit is even
3: Sum of digits divisible by 3
4: Last two digits divisible by 4
5: Last digit is 0 or 5
6: Divisible by both 2 and 3
7: Double the last digit, subtract from rest
8: Last 3 digits divisible by 8
9: Sum of digits divisible by 9
10: Last digit is 0
11: Alternating sum of digits divisible by 11
12: Divisible by 3 and 4
13-20: Use long division or patterns from remainder arithmetic
5. Even and Odd Properties
Even ± Even = Even
Odd ± Odd = Even
Even ± Odd = Odd
Even × Even = Even
Odd × Odd = Odd
Even × Odd = Even
6. Prime and Composite Numbers
Prime Numbers:
Greater than 1
Exactly two factors: 1 and itself
Examples: 2, 3, 5, 7, 11, 13...
Composite Numbers:
Have more than two factors
Example: 4, 6, 8, 9, 10
Co-prime Numbers:
Two numbers with HCF = 1
Example: 15 and 28
7. Factors and Multiples
Factor
A number that divides another without leaving a remainder
Factors of 12: 1, 2, 3, 4, 6, 12
Multiple
Product of a number and an integer
Multiples of 3: 3, 6, 9, 12...
Prime Factorization
Breaking a number into product of prime numbers
Example: 60 = 2^2 × 3 × 5
HCF & LCM
HCF (GCD): Greatest common divisor
LCM: Lowest number divisible by both
Formula: HCF × LCM = Product of the numbers
Example:
Find HCF and LCM of 24 and 36
24 = 2^3 × 3
36 = 2^2 × 3^2
HCF = 2^2 × 3 = 12
LCM = 2^3 × 3^2 = 72
8. Remainders and Modular Arithmetic
Euclidean Division:
a = bq + r
r is remainder (0 ≤ r < b)
Fermat's Little Theorem:
a^(p-1) ≡ 1 (mod p), if p is prime and does not divide a
Euler’s Theorem:
a^(ϕ(m)) ≡ 1 (mod m), if a and m are co-prime
Remainder Trick:
Find remainder when 7^100 is divided by 4
7 mod 4 = 3
3^n mod 4 pattern = 3, 1...
100 is even → Remainder = 1
9. Number Base Systems
Binary System:
Base 2: Only 0 and 1
Used in computing
Octal System:
Base 8: Digits 0 to 7
Decimal System:
Base 10: Digits 0 to 9
Hexadecimal:
Base 16: 0-9 and A-F (10-15)
Example:
Convert (1101)2 to decimal= 1Ò8 + 1Ò4 + 0Ò2 + 1Ò1 = 13
10. Units Digit and Last Digit Questions
Units Digit of Exponents:
Find cyclicity:
Last digit of 7^1 = 7
7^2 = 49 (9), 7^3 = 343 (3), 7^4 = 2401 (1)
Pattern: 7, 9, 3, 1 (repeats every 4)
Example:
7^45 → 45 mod 4 = 1 → First in cycle = 7
11. Digital Root & Casting Out Nines
Digital Root:
Sum digits repeatedly until one digit
Example: 9876 → 9+8+7+6 = 30 → 3+0 = 3
Use:
Quick calculation check
Match digital roots in multiple-choice
12. Special Numbers
Perfect Numbers:
Sum of proper divisors = Number
Examples: 6, 28
Amicable Numbers:
Sum of proper divisors of A = B and vice versa
Example: 220 and 284
Armstrong Numbers:
Sum of digits^n = Number
Example: 153 = 1^3 + 5^3 + 3^3
Palindromes:
Same forward and backward
121, 1331
13. Fast Mental Math Tricks
Multiplying by 11:
52 × 11: 5 (5+2) 2 = 572
Square of numbers ending in 5:
35^2 = 3×4 = 12, then append 25 → 1225
Squaring numbers near 100:
104^2 = (100+4)^2 = 100^2 + 2×100×4 + 16 = 10816
14. Word Problems & Exam Tricks
Problem 1:
Find smallest number that leaves remainder 2 when divided by 5 and remainder 4 when divided by 7.
Check multiples of 7 + 4: 11, 18, 25... Find which leaves 2 mod 5: 24
Problem 2:
What number is divisible by 8, 9, and 12?
LCM(8,9,12) = 72
Problem 3:
Remainder of 9^23 divided by 5
Pattern: 4, 1, 4, 1... → 23 mod 2 = 1 → Answer = 4
15. Common Mistakes
Confusing remainder with quotient
Not applying power cycles
Using wrong base in number conversions
Forgetting HCF×LCM = product of numbers
16. Practice Set
Convert 45 to binary.
Find HCF and LCM of 18 and 30.
Unit digit of 19^57?
Sum of first 25 odd numbers?
What is digital root of 4982?
Answers:
101101
HCF = 6, LCM = 90
Cycle: 9,1 → 57 mod 2 = 1 → 9
n^2 = 625
4+9+8+2 = 23 → 2+3 = 5
17. Final Summary & Preparation Plan
Day-Wise Plan (15 Days):
Day 1-2: Number classification and divisibility
Day 3-4: HCF, LCM, Prime factorization
Day 5-6: Remainders and Euler’s Theorem
Day 7-8: Number bases and binary conversions
Day 9-10: Digital roots and special numbers
Day 11-13: Word problems, unit digits
Day 14: Full-length practice test
Day 15: Analyze errors and revise formulas
With consistent practice, Number System becomes a high-scoring and reliable section. Keep revising, use mental math, and solve problems daily to strengthen your command over this crucial topic.
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