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Topic 9

Part of the NABE (Pakistan) study roadmap. Subject Specific topic subjec-009 of Subject Specific.

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Number System

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Number System — Key Facts for NABE (Pakistan)

  • Natural Numbers: 1, 2, 3, 4, … | Whole Numbers: 0, 1, 2, 3, …
  • Integers: …-3, -2, -1, 0, 1, 2, 3, … | Rational: p/q where q ≠ 0
  • Divisibility: Key rules — 2 (even), 3 (digit sum), 5 (ends 0/5), 9 (digit sum), 10 (ends 0)
  • HCF × LCM = Product of two numbers (for any two positive integers)
  • ⚡ Exam tip: Number of factors from prime factorization: if N = p^a × q^b, then factors = (a+1)(b+1)

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Number System — NABE (Pakistan) Study Guide

Classification of Numbers

Numbers
├── Natural Numbers: 1, 2, 3, ... (counting numbers)
├── Whole Numbers: 0, 1, 2, 3, ... (natural + zero)
├── Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
├── Even Numbers: Divisible by 2 (0, 2, 4, 6, ...)
├── Odd Numbers: Not divisible by 2 (1, 3, 5, 7, ...)
├── Prime Numbers: Divisible only by 1 and itself (2, 3, 5, 7, 11, ...)
├── Composite Numbers: Have other factors (4, 6, 8, 9, 10, ...)
└── Rational Numbers: Can be expressed as p/q where q ≠ 0

Divisibility Rules

DivisorRule
2Last digit is even (0, 2, 4, 6, 8)
3Sum of digits is divisible by 3
4Last two digits form a number divisible by 4
5Last digit is 0 or 5
6Divisible by 2 AND 3
8Last three digits form a number divisible by 8
9Sum of digits is divisible by 9
10Last digit is 0

HCF and LCM

HCF (GCD): Highest Common Factor — largest number that divides both

LCM: Lowest Common Multiple — smallest number divisible by both

Key Formula: For any two positive integers a and b:

HCF(a, b) × LCM(a, b) = a × b

Example: Find HCF and LCM of 12 and 18

  • 12 = 2² × 3
  • 18 = 2 × 3²
  • HCF = 2¹ × 3¹ = 6
  • LCM = 2² × 3² = 36
  • Check: 6 × 36 = 12 × 18 = 216 ✓

NABE Exam Pattern

Common question types:

  1. Divisibility and factors
  2. HCF and LCM calculations
  3. Prime factorization
  4. Number of factors and sum of factors
  5. Finding remainders

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Number System — Comprehensive NABE (Pakistan) Notes

Detailed Theory

1. Number Classification — Expanded

Natural Numbers (N): 1, 2, 3, 4, …

  • Subset: Even (2, 4, 6, …), Odd (1, 3, 5, …)

Whole Numbers (W): 0, 1, 2, 3, …

Integers (Z): All whole numbers and their negatives

  • Z⁺ = Positive integers = {1, 2, 3, …}
  • Z⁻ = Negative integers = {…-3, -2, -1}
  • Z₀ = {…-2, -1, 0, 1, 2, …}

Rational Numbers (Q): p/q where p, q are integers, q ≠ 0

  • Includes all fractions and integers
  • Decimal expansion: terminating or repeating

Irrational Numbers: Cannot be expressed as p/q

  • Examples: √2, π, e, √3
  • Decimal expansion: non-terminating, non-repeating

Real Numbers (R): Rational + Irrational

2. Prime Numbers — Complete Analysis

Definition: A prime number has exactly two distinct factors: 1 and itself.

First 20 Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71

Properties of Prime Numbers:

  • 2 is the only even prime
  • All primes greater than 3 are of the form 6k ± 1
  • There are infinitely many primes (Euclid’s proof)
  • No formula generates all primes

Prime Testing:

  • To check if n is prime, test divisibility by primes ≤ √n
  • Example: Is 97 prime? Check primes ≤ √97 ≈ 9.8 → 2, 3, 5, 7
  • 97 not divisible by any → 97 is prime

Sieve of Eratosthenes: Method to find all primes up to n

  1. List numbers 2 to n
  2. Mark 2, cross out all multiples of 2
  3. Next unmarked number is prime, cross out its multiples
  4. Repeat until √n

3. Co-Prime Numbers

Definition: Two numbers are co-prime if their HCF = 1

  • Examples: (8, 15), (7, 9), (14, 15)
  • Note: Co-prime numbers need not be primes individually

Properties:

  • If a and b are co-prime: LCM(a, b) = a × b
  • φ(n) = count of numbers less than n and co-prime to n (Euler’s totient)

4. HCF and LCM — Methods

Prime Factorization Method:

HCF: Take MINIMUM powers of common primes
LCM: Take MAXIMUM powers of all primes

Example: Find HCF and LCM of 60, 72, and 120

  • 60 = 2² × 3¹ × 5¹
  • 72 = 2³ × 3²
  • 120 = 2³ × 3¹ × 5¹
  • HCF = 2² × 3¹ = 12
  • LCM = 2³ × 3² × 5¹ = 360

Division Method:

  • LCM: Divide successively by primes until all quotients are 1
  • HCF: Divide larger by smaller, then remainder by previous divisor until remainder = 0

5. Number of Factors (Divisors)

If N = p^a × q^b × r^c, then:

Number of Factors (d) = (a+1)(b+1)(c+1)
Sum of Factors (σ) = [(p^(a+1)-1)/(p-1)] × [(q^(b+1)-1)/(q-1)] × ...

Example: Find number of factors of 180

  • 180 = 2² × 3² × 5¹
  • d(180) = (2+1)(2+1)(1+1) = 3 × 3 × 2 = 18 factors

Perfect Numbers: Sum of proper divisors equals the number

  • Example: 6 (1+2+3=6), 28 (1+2+4+7+14=28)

6. Factorial Notation

n! = 1 × 2 × 3 × … × n

Highest Power of Prime p in n!:

v_p(n!) = ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + ...

Example: Highest power of 3 in 10!

  • v₃(10!) = ⌊10/3⌋ + ⌊10/9⌋ = 3 + 1 = 4
  • 10! = 3628800, which is divisible by 3⁴ = 81 but not 3⁵ = 243

7. Cyclicity and Last Digits

Last digit patterns cycle:

  • 2^1=2, 2^2=4, 2^3=8, 2^4=6, 2^5=2… cycle: 2,4,8,6
  • 3^1=3, 3^2=9, 3^3=7, 3^4=1, 3^5=3… cycle: 3,9,7,1
  • 7^1=7, 7^2=9, 7^3=3, 7^4=1, 7^5=7… cycle: 7,9,3,1

To find last digit of a^b: Find remainder of b divided by cycle length, use that power.

8. Congruences (Basics)

Definition: a ≡ b (mod m) means m divides (a-b)

Properties:

  • a ≡ a (mod m) [reflexive]
  • If a ≡ b, then b ≡ a (mod m) [symmetric]
  • If a ≡ b and b ≡ c, then a ≡ c (mod m) [transitive]

Application: Finding remainders

  • Example: Find remainder when 2^100 is divided by 3
  • 2 ≡ -1 (mod 3)
  • 2^100 ≡ (-1)^100 ≡ 1 (mod 3)
  • Remainder = 1

If numbers form an AP with common difference d:

  • Sum of n terms: n/2 × (first + last) = n/2 × [2a + (n-1)d]
  • Sum of first n natural numbers: n(n+1)/2

10. Common Mistakes to Avoid

  1. 1 is not prime: 1 has only one factor, primes need exactly two
  2. HCF × LCM: Only applies to two numbers (not three)
  3. Divisibility check: Test all applicable rules
  4. Proper factors: Exclude the number itself
  5. Negative numbers: Be careful with divisibility rules

Practice Questions for NABE

  1. Find the HCF and LCM of 72, 120, and 156 using prime factorization.
  2. How many factors does 360 have? List them.
  3. Find the highest power of 5 in 100!
  4. If the LCM of two numbers is 180 and their HCF is 15, and one number is 45, find the other.
  5. Prove that the product of three consecutive integers is always divisible by 6.

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