Number system: Quantitative Aptitude Notes for PTE

📚 PTE Quantitative Aptitude – Number System

(Study Notes for Quick Revision)

1️⃣ Simple Explanation

A number system is a set of numbers and the rules for manipulating them. In the PTE exam we mainly deal with:

System	Definition	Examples
Natural numbers	Counting numbers starting from 1.	1, 2, 3, …
Whole numbers	Natural numbers + 0.	0, 1, 2, …
Integers	Whole numbers + their negatives.	…‑3, ‑2, ‑1, 0, 1, 2, …
Rational numbers	Numbers that can be expressed as a fraction `p/q` where `p, q` are integers and `q ≠ 0`.	½, 4, –3/7, 0.75
Irrational numbers	Numbers that cannot be expressed as a ratio of integers (non‑terminating, non‑repeating decimal).	√2, π, e
Real numbers	All rational + irrational numbers.	0, 1/3, √5
Complex numbers	Numbers of the form `a + bi` where `i² = –1`.	3 + 4i

In PTE you’ll mostly need to identify which category a number belongs to and use simple properties (divisibility, parity, etc.) to solve problems.

2️⃣ Key Points & Rules

Topic	Key Point	Quick Test
Parity	Even numbers are divisible by 2; odd numbers are not.	Write `n ÷ 2`. If remainder 0 → even.
Divisibility Tests	2,3,4,5,6,7,8,9,10,11,12,13,15,20,25,30,36,40,50,60,70,100.	Remember the standard rules:
	2	last digit even
	3	sum of digits divisible by 3
	4	last two digits divisible by 4
	5	last digit 0 or 5
	6	divisible by both 2 & 3
	7	Rule of 2×last digit (subtract 2×last digit from remaining number; repeat)
	8	last three digits divisible by 8
	9	sum of digits divisible by 9
	10	last digit 0
	11	alternating sum of digits divisible by 11
	12	divisible by both 3 & 4
	13	Rule of 4×last digit (add 4×last digit to remaining number; repeat)
	15	divisible by both 3 & 5
	20	last two digits 00,20,40,60,80
	25	last two digits 00,25,50,75
	30	divisible by both 3 & 10
	36	divisible by both 4 & 9
	40	last two digits 00,20,40,60,80
	50	last digit 0 or 5 with second last digit 0
	60	divisible by both 3 & 20
	70	divisible by both 5 & 14
	100	last two digits 00
Prime Numbers	A prime has exactly two distinct positive divisors: 1 and itself.	Test divisibility up to √n.
Composite Numbers	Not prime; has more than two divisors.	Any number >1 that isn’t prime.
Perfect Squares	`n = k²` (e.g., 1,4,9,16).	Last digit 0,1,4,5,6,9; check square root.
Perfect Cubes	`n = k³` (e.g., 1,8,27,64).	Last digit 0,1,8,7,6,5,4,3,2,9; check cube root.
Factorial & Exponentials	`n! = 1×2×…×n`. Powers `a^b`.	Use prime factorization for large numbers.
Number Representation	Integer part + fractional part.	Convert fractions to decimals or vice‑versa.
Decimal, Binary, Octal, Hexadecimal	Representations in base‑10, base‑2, base‑8, base‑16.	Understand conversion rules (e.g., 1010₂ = 10₁₀).

3️⃣ Worked Example

Problem (PTE‑style)
Find how many integers between 100 and 200 (inclusive) are divisible by 12 but not by 9.

Step‑by‑Step Solution

Find multiples of 12 in the range.
- Smallest multiple of 12 ≥ 100:
  [ \lceil 100/12 \rceil = 9 \Rightarrow 9 \times 12 = 108 ]
- Largest multiple of 12 ≤ 200:
  [ \lfloor 200/12 \rfloor = 16 \Rightarrow 16 \times 12 = 192 ]
- Count = 16 – 9 + 1 = 8 multiples: 108, 120, 132, 144, 156, 168, 180, 192.
Filter out multiples of 9.
- A number divisible by both 12 and 9 is divisible by LCM(12,9) = 36.
- Find multiples of 36 in the range:
  [ \lceil 100/36 \rceil = 3 \Rightarrow 3 \times 36 = 108 ] [ \lfloor 200/