Number System

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Number System — Key Facts for GAT Pakistan

Types of Numbers:

Type	Definition	Example
Natural Numbers (N)	Counting numbers starting from 1	1, 2, 3, 4, …
Whole Numbers (W)	Natural numbers + 0	0, 1, 2, 3, …
Integers (Z)	Positive, negative, and zero	…, -2, -1, 0, 1, 2, …
Rational Numbers (Q)	Can be expressed as p/q where q≠0	1/2, 3/4, -5/6, 0.333
Irrational Numbers	Cannot be expressed as p/q	√2, π, e
Real Numbers (R)	All rational + irrational	All numbers on number line

Even and Odd Numbers:

• Even: Divisible by 2 → 2, 4, 6, 8, 10
• Odd: Not divisible by 2 → 1, 3, 5, 7, 9
• Properties:
  • Even + Even = Even
  • Odd + Odd = Even
  • Even + Odd = Odd
  • Even × Even = Even
  • Odd × Odd = Odd
  • Even × Odd = Even

Prime Numbers:

• Numbers divisible only by 1 and themselves
• First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
• Note: 1 is NOT a prime number
• Only even prime: 2

⚡ GAT Exam Tip: For prime number questions, remember that 2 is the only even prime. All other primes are odd.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Number System — Detailed Study Guide

Divisibility Rules

Divisor	Rule	Example
2	Last digit even (0, 2, 4, 6, 8)	124 ✓, 137 ✗
3	Sum of digits divisible by 3	126 → 1+2+6=9 ✓
4	Last two digits divisible by 4	124 ✓ (24÷4=6)
5	Last digit 0 or 5	135 ✓
6	Divisible by both 2 and 3	126 ✓
7	Double last digit, subtract from rest	203 → 20-6=14 ✓
8	Last three digits divisible by 8	1248 ✓
9	Sum of digits divisible by 9	126 → 1+2+6=9 ✓
10	Last digit 0	120 ✓
11	(Sum of odd positions) - (Sum of even positions) divisible by 11	2728: (2+7) - (7+8) = 9-15 = -6 ✗

GAT Worked Example:

Question: Is 1734 divisible by 3?
Solution:
Sum of digits = 1 + 7 + 3 + 4 = 15
15 ÷ 3 = 5 (exact)
Answer: Yes, 1734 is divisible by 3

⚡ Common Mistake: Students often forget that divisibility by 6 requires BOTH 2 and 3 conditions to be satisfied.

LCM and HCF

HCF (GCD - Greatest Common Divisor):

• Largest number that divides both numbers
• Method 1: Prime factorization
  • Find prime factors of each number
  • Take common factors with lowest power
• Method 2: Euclidean Algorithm
  • Divide larger by smaller
  • Replace larger with remainder
  • Continue until remainder = 0

LCM (Least Common Multiple):

• Smallest number divisible by both numbers
• Method 1: Prime factorization
  • Take highest power of each prime
• Method 2: LCM × HCF = Product of two numbers

Worked Examples:

Example 1: Find HCF and LCM of 24 and 36

Prime factorization:
24 = 2³ × 3
36 = 2² × 3²

HCF = 2² × 3 = 12
LCM = 2³ × 3² = 72

Verification: 24 × 36 = 864
             HCF × LCM = 12 × 72 = 864 ✓

Example 2: Using Euclidean Algorithm
HCF of 72 and 48:
72 ÷ 48 = 1 remainder 24
48 ÷ 24 = 2 remainder 0
HCF = 24 ✓

⚡ GAT PYQ: “The HCF of 48, 72, and 96 is: (a) 24 (b) 12 (c) 48 (d) 96” → Answer: (a) 24

Fractions and Decimals

Types of Decimals:

Type	Example	Conversion
Terminating	0.25, 0.125	Finite digits after decimal
Recurring	0.333…	Repeating pattern
Non-terminating non-recurring	√2 = 1.414…	No pattern

Fraction Conversions:

1/2 = 0.5
1/3 = 0.333... = 0.3̄
1/4 = 0.25
1/5 = 0.2
1/6 = 0.1666... = 0.16̄
1/8 = 0.125
1/9 = 0.111... = 0.1̄
1/10 = 0.1

To convert fraction to decimal: Divide numerator by denominator To convert decimal to fraction: Count decimal places, put over 10^n

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Number System — Complete Notes for GAT

Surds and Indices

Laws of Indices:

Law	Formula	Example
Product	a^m × a^n = a^(m+n)	2³ × 2⁴ = 2⁷
Quotient	a^m ÷ a^n = a^(m-n)	3⁵ ÷ 3² = 3³
Power of power	(a^m)^n = a^(mn)	(2²)³ = 2⁶
Zero exponent	a^0 = 1 (where a≠0)	5⁰ = 1
Negative exponent	a^(-n) = 1/a^n	2^(-3) = 1/8
Fractional exponent	a^(m/n) = n√(a^m)	8^(2/3) = ³√(8²) = 4

Laws of Surds:

Law	Formula	Example
Multiplication	√a × √b = √(ab)	√3 × √5 = √15
Division	√a ÷ √b = √(a/b)	√15 ÷ √3 = √5
Rationalization	√a × √a = a	√5 × √5 = 5
Conjugate	(√a + √b)(√a - √b) = a - b	(√7+√3)(√7-√3) = 7-3=4

Worked Examples:

Example 1: Simplify ³√72
72 = 8 × 9 = 2³ × 3²
³√72 = ³√(2³ × 3²) = 2 × ³√9 = 2³√9

Example 2: Simplify (√3 + √2)(√3 - √2)
= (√3)² - (√2)²
= 3 - 2 = 1 ✓ (using difference of squares)

Example 3: Simplify 2√3 + 5√3 - 3√3
= (2 + 5 - 3)√3
= 4√3

⚡ Important for GAT: Always rationalize denominators when simplifying expressions with surds.

Number Theory Problems

Finding Number of Factors:

Example: Find the number of positive factors of 360

360 = 2³ × 3² × 5¹

Number of factors = (3+1)(2+1)(1+1)
                  = 4 × 3 × 2
                  = 24

The 24 factors are: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18,
                    20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360

Perfect Numbers:

• A number equal to the sum of its proper divisors
• Example: 6 = 1 + 2 + 3 (proper divisors)
• Example: 28 = 1 + 2 + 4 + 7 + 14
• Next perfect number: 496, then 8128

Prime Factorization Applications:

Question: Find the largest 4-digit number divisible by 12, 15, and 18.

Solution:
12 = 2² × 3
15 = 3 × 5
18 = 2 × 3²

LCM = 2² × 3² × 5 = 180

Largest 4-digit number = 9999
9999 ÷ 180 = 55 remainder 99
9999 - 99 = 9900

Answer: 9900

GAT-Style Practice Questions

1. If a number leaves remainder 3 when divided by 5, and remainder 5 when
   divided by 7, what is the least such number?
   (a) 33 (b) 38 (c) 45 (d) 48
   
   Answer: (a) 33
   Solution: 33 ÷ 5 = 6 remainder 3 ✓
             33 ÷ 7 = 4 remainder 5 ✓

2. The difference between the largest and smallest 3-digit numbers
   divisible by 17 is:
   (a) 884 (b) 867 (c) 952 (d) 999
   
   Answer: (b) 867
   Solution: Smallest 3-digit = 102 (17×6)
             Largest 3-digit = 986 (17×58)
             Difference = 986 - 102 = 884
             (Wait, let me recalculate)
             Smallest 3-digit divisible by 17: 17×6 = 102 ✓
             Largest 3-digit divisible by 17: 17×58 = 986
             17×59 = 1003 (4-digit)
             Difference = 986 - 102 = 884
             Answer should be (a) 884

3. How many prime numbers are between 1 and 50?
   (a) 14 (b) 15 (c) 16 (d) 17
   
   Answer: (b) 15
   Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 = 15 primes

⚡ GAT Strategy: For divisibility questions, first check the conditions. For remainder questions, use the Chinese Remainder Theorem approach for multiple divisors.

Unit Digit Problems

Finding Unit Digits of Powers:

Base Unit Digit	Pattern Cycle	Example
0	Always 0	0, 0, 0, …
1	Always 1	1, 1, 1, …
2	2, 4, 8, 6	2^15 ends in 2^3 = 8
3	3, 9, 7, 1	3^15 ends in 3^3 = 7
4	4, 6	4^15 ends in 4
5	Always 5	5, 5, 5, …
6	Always 6	6, 6, 6, …
7	7, 9, 3, 1	7^15 ends in 7^3 = 3
8	8, 4, 2, 6	8^15 ends in 8^3 = 2
9	9, 1	9^15 ends in 9

Worked Example:
Find the unit digit of 7^23

Solution:
Cycle for 7: 7, 9, 3, 1 (every 4)
23 ÷ 4 = 5 remainder 3
Position 3 in cycle = 3

Answer: Unit digit = 3

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