Number system: Quantitative Aptitude Notes for Madhya Pradesh PSC

📚 Number System – Quantitative Aptitude (Madhya Pradesh PSC)

Goal:
Master the fundamentals of the number system, recognise different kinds of numbers, and solve quick‑fire problems that frequently appear in the Madhya Pradesh Public Service Commission (MPPSC) exams.

1️⃣ Simple Explanation

The Number System deals with the different kinds of numbers and the operations that can be performed on them.
In MPPSC quantitative aptitude, you’ll most often encounter:

Category	Symbol	Typical Questions
Natural Numbers	1, 2, 3, …	“How many natural numbers ≤ 200 are divisible by 5?”
Whole Numbers	0, 1, 2, …	“How many whole numbers ≤ 100 are square numbers?”
Integers	…, –3, –2, –1, 0, 1, 2, 3, …	“What is the sum of the first 10 negative integers?”
Rational Numbers	Fractions & terminating/recurring decimals	“Express 0.75 as a fraction.”
Irrational Numbers	√2, π, e	“Is √2 rational? Explain.”
Real Numbers	All numbers that can be written on a number line	“Which of the following are real numbers?”
Complex Numbers	a + bi	“Compute (3 + 2i) + (1 – 4i).”

Key Idea:
In exam problems you’ll usually need to identify the type of number, convert between forms, or apply basic properties (e.g., “If a and b are integers, then a² + b² is even/odd?”).

2️⃣ Key Points (Bullet‑style)

#	Point	Why It Matters (MPPSC)
1	Place Value – Units, Tens, Hundreds, etc.	Quickly find digit sums, check divisibility.
2	Common Divisibility Rules	2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 15, 16, 20, 25, 50, 100.
3	Prime & Composite Numbers	Solve “how many primes up to 50?” or “factorise 84”.
4	Perfect Squares / Cubes	“How many perfect squares ≤ 200?”
5	Factorisation & GCD/LCD	Essential for fraction simplification and ratio problems.
6	Rational vs Irrational	“Is 3.1416 rational? Why?”
7	Scientific Notation	Handle very large/small numbers quickly.
8	Repetend Length	Identify repeating decimals; e.g., 1/7 = 0.(142857).
9	Conversion between Fraction & Decimal	“Write 5/8 as a decimal.”
10	Number Properties	Parity (even/odd), sign, range, etc.
11	Arithmetic Operations	Addition, subtraction, multiplication, division with numbers of any type.
12	Modular Arithmetic	Useful for “last digit of 7^100” or “remainder when 1234 is divided by 7”.
13	Number Patterns	Recognise arithmetic, geometric, quadratic, or Fibonacci sequences.

3️⃣ Worked Example

Question (MPPSC style)

How many integers between 1 and 1000 (inclusive) are divisible by 7 but not divisible by 5?

Step‑by‑Step Solution

Step	Calculation	Explanation
1	Find numbers divisible by 7: ⌊1000 ÷ 7⌋ = 142	142 multiples of 7 up to 1000
2	Find numbers divisible by 35 (LCM of 7 & 5): ⌊1000 ÷ 35⌋ = 28	These are divisible by both 7 and 5
3	Subtract: 142 – 28 = 114	Numbers divisible by 7 and not by 5
Answer	114	114 integers satisfy the condition

Quick Check (Cross‑Verification)

Multiples of 7: 7, 14, 21, …, 994.
Every 35th of these will also be a multiple of 5 (35, 70, 105, …).
Removing them gives the correct count.

4️⃣ Quick Revision Summary (Flash‑card Style)

Topic	Quick Tip
Place Value	Digit × place value gives the number.
Divisibility by 3/9	Sum of digits divisible by 3/9.
Divisibility by 4	Last two digits divisible by 4.
Divisibility by 5	Last digit 0 or 5.
Divisibility by 7	Use 2×(last digit) subtraction trick or 10ⁿ mod 7.
Divisibility by 11	Alternating sum of digits ≡ 0 (mod 11).
Prime Test	Check divisibility up to √n.
Perfect Squares	n² where n integer.
Conversion	0.75 = 75/100 = 3/4.
Repeating Decimals	1/3 = 0.(3); 1/6 = 0.1(6).
Scientific Notation	6,400 = 6.4 × 10³.
LCM/GCD	Use prime factorisation or Euclid's algorithm.
Modular Arithmetic	a ≡ b (mod m) → same remainder when divided by m.
Patterns	Identify if sequence is arithmetic (difference constant) or geometric (ratio constant).

📌 Final Tips for MPPSC

Practice Divisibility Rules – They’re exam time‑