Since your platform already supports KaTeX + Markdown rendering (as you mentioned in your renderer pipeline), the lesson should be written in pure Markdown with LaTeX blocks so KaTeX can render it cleanly.

Below is Lesson 1 rewritten in KaTeX-compatible Markdown (no HTML, only MD + math).

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CSAT Number System

Lesson 1: Understanding Numbers

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1. What is a Number System?

A number system is a way of representing and classifying numbers based on their properties.

Understanding the classification of numbers helps solve many CSAT questions without heavy calculations.

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2. Types of Numbers

2.1 Natural Numbers

Natural numbers are counting numbers starting from 1.

$$\mathbb{N} = {1,2,3,4,5,6,7,\dots}$$

Properties

• Smallest natural number = $1$
• No largest natural number
• Infinite set

Example

$$3,\ 8,\ 15,\ 102$$

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2.2 Whole Numbers

Whole numbers include zero along with natural numbers.

$$\mathbb{W} = {0,1,2,3,4,5,\dots}$$

Property

Smallest whole number

$$0$$

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2.3 Integers

Integers include negative numbers, zero, and positive numbers.

$$\mathbb{Z} = {-3,-2,-1,0,1,2,3,\dots}$$

Examples

$$-5,\ -2,\ 0,\ 7$$

Applications include

• temperature
• profit and loss
• elevation

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2.4 Rational Numbers

A number is rational if it can be written as

$$\frac{p}{q}$$

where

$$p,q \in \mathbb{Z}, \quad q \ne 0$$

Examples

$$\frac{1}{2},\quad \frac{5}{3},\quad -4,\quad \frac{7}{1}$$

Decimals of rational numbers are

• terminating
• repeating

Examples

$$0.25,\quad 0.3333\dots$$

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2.5 Irrational Numbers

Numbers that cannot be expressed as $\frac{p}{q}$.

Their decimal expansions are

• non-terminating
• non-repeating

Examples

$$\sqrt{2},\quad \sqrt{3},\quad \pi$$

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2.6 Real Numbers

All numbers on the number line are real numbers.

$$\mathbb{R} = \text{Rational Numbers} \cup \text{Irrational Numbers}$$

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3. Even and Odd Numbers

Even Numbers

Numbers divisible by $2$.

Examples

$$2,\ 4,\ 6,\ 8,\ 10$$

Quick identification rule

If the last digit is

$$0,2,4,6,8$$

the number is even.

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Odd Numbers

Numbers not divisible by $2$.

Examples

$$1,\ 3,\ 5,\ 7,\ 9$$

Odd numbers end in

$$1,3,5,7,9$$

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4. Even–Odd Operation Rules

Example

$$\text{Odd} \times \text{Odd} \times \text{Even} = \text{Even}$$

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5. Prime Numbers

A prime number has exactly two distinct factors:

$$1 \quad \text{and} \quad p$$

Examples

$$2,\ 3,\ 5,\ 7,\ 11,\ 13$$

Important property

$$2$$

is the only even prime number.

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6. Composite Numbers

Numbers having more than two factors.

Examples

$$4,\ 6,\ 8,\ 9,\ 10$$

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7. Factors and Multiples

Factors

A factor divides a number exactly.

Example: Factors of $12$

$$1,2,3,4,6,12$$

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Multiples

Multiples are obtained by multiplying a number.

Multiples of $5$

$$5,10,15,20,25,\dots$$

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8. CSAT Quick Observations

Trick 1

Product of two consecutive integers

$$n(n+1)$$

Always even.

Example

$$7 \times 8 = 56$$

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Trick 2

Product of three consecutive integers

$$n(n+1)(n+2)$$

Always divisible by

$$6$$

Example

$$4 \times 5 \times 6 = 120$$

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Trick 3

Difference of squares of consecutive numbers

$$(n+1)^2 - n^2$$

Simplifies to

$$2n + 1$$

Which is always odd.

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9. Concept Check

Question 1

Which of the following is irrational?

A. $3/5$ B. $\sqrt{7}$ C. $0.25$ D. $5$

Answer

$$\sqrt{7}$$

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Question 2

The only even prime number is

$$2$$

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Question 3

Evaluate

$$\text{Odd} \times \text{Odd} \times \text{Even}$$

Answer

$$\text{Even}$$

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Lesson Summary

• Natural numbers start from $1$
• Whole numbers start from $0$
• Integers include negative numbers
• Rational numbers can be written as $\frac{p}{q}$
• Irrational numbers cannot be written as $\frac{p}{q}$
• Prime numbers have exactly two factors
• Even–odd rules allow instant CSAT solutions

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If you'd like, I can also create Lesson 2 (Divisibility Rules with 2-second shortcuts) in the same KaTeX Markdown format, which is actually one of the most powerful CSAT scoring areas.