Number System

Overview

The Number System forms the bedrock of primary mathematics and carries significant weightage in OTET Paper I. Questions from this topic test your understanding of how numbers are built, classified, and related to each other. Mastery here directly supports performance in fractions, decimals, LCM-HCF, and word problems.

For the exam, you must be comfortable with the hierarchy of number types (natural → whole → integers), place value concepts up to crores, and the relationship between factors and multiples. Most questions are straightforward calculations or concept-based MCQs, but careless errors in place value or confusing factors with multiples cost marks.

Focus on building a clear mental model of number classification, quick recall of divisibility rules, and confident handling of negative integers. These skills appear repeatedly across the mathematics section.

Key Concepts

• **Natural Numbers**: Counting numbers starting from 1 (i.e., 1, 2, 3, 4, ...). Zero is not a natural number.

• **Whole Numbers**: Natural numbers plus zero (i.e., 0, 1, 2, 3, ...). Every natural number is a whole number, but zero is only a whole number.

• **Integers**: Whole numbers plus their negatives (..., -3, -2, -1, 0, 1, 2, 3, ...). Integers include positive numbers, negative numbers, and zero.

• **Place Value vs Face Value**: Place value depends on the position of a digit (e.g., 5 in 3527 has place value 500), while face value is the digit itself (face value of 5 is always 5).

• **Factors**: Numbers that divide a given number exactly without leaving a remainder. Every number has at least two factors: 1 and itself.

• **Multiples**: Numbers obtained by multiplying a given number by natural numbers. A number has infinitely many multiples.

• **Prime Numbers**: Numbers greater than 1 with exactly two factors (1 and the number itself). Examples: 2, 3, 5, 7, 11, 13.

• **Composite Numbers**: Numbers greater than 1 with more than two factors. The number 1 is neither prime nor composite.

Formulas / Key Facts

| Concept | Key Fact | |---------|----------| | Smallest natural number | 1 | | Smallest whole number | 0 | | Smallest prime number | 2 (also the only even prime) | | Number of factors of a prime | Exactly 2 | | 1 is | Neither prime nor composite | | Place value of digit d at position p | d × 10^(p-1), counting from right starting at position 1 | | Sum of first n natural numbers | n(n+1)/2 | | Product of two numbers | LCM × HCF |

**Divisibility Rules (must memorize):**

• By 2: Last digit is 0, 2, 4, 6, or 8
• By 3: Sum of digits is divisible by 3
• By 4: Last two digits form a number divisible by 4
• By 5: Last digit is 0 or 5
• By 6: Divisible by both 2 and 3
• By 9: Sum of digits is divisible by 9
• By 10: Last digit is 0
• By 11: Difference of sum of digits at odd and even places is 0 or divisible by 11

Worked Examples

**Example 1: Place Value** *Find the difference between the place value and face value of 7 in 47832.*

Step 1: Identify position of 7 — it is in the thousands place. Step 2: Place value = 7 × 1000 = 7000 Step 3: Face value = 7 Step 4: Difference = 7000 - 7 = **6993**

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**Example 2: Finding All Factors** *List all factors of 36.*

Step 1: Start dividing from 1 and find pairs.

• 36 ÷ 1 = 36 → factors: 1, 36
• 36 ÷ 2 = 18 → factors: 2, 18
• 36 ÷ 3 = 12 → factors: 3, 12
• 36 ÷ 4 = 9 → factors: 4, 9
• 36 ÷ 6 = 6 → factors: 6

Step 2: Factors of 36 = **1, 2, 3, 4, 6, 9, 12, 18, 36** (9 factors)

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**Example 3: Integer Operations** *Simplify: (-15) + 8 + (-3) - (-10)*

Step 1: Rewrite subtraction as addition: (-15) + 8 + (-3) + 10 Step 2: Group positives and negatives:

• Positives: 8 + 10 = 18
• Negatives: (-15) + (-3) = -18

Step 3: Result = 18 + (-18) = **0**

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**Example 4: Divisibility Check** *Is 5765 divisible by 11?*

Step 1: Digits at odd places (from right): 5, 7 → Sum = 12 Step 2: Digits at even places: 6, 5 → Sum = 11 Step 3: Difference = 12 - 11 = 1 Step 4: Since 1 is not divisible by 11, **5765 is not divisible by 11**.

Common Mistakes

• **Confusing factors and multiples** → Remember: Factors divide into the number (are smaller or equal); multiples are obtained by multiplying (are larger or equal). "Factors are few, multiples are many."

• **Treating 1 as prime** → 1 has only one factor (itself), so it fails the "exactly two factors" rule. Always exclude 1 from prime lists.

• **Forgetting zero in whole numbers** → Students often say "whole numbers start from 1." Whole numbers start from 0; natural numbers start from 1.

• **Place value errors with zeros** → The digit 0 in 3045 at the hundreds place has place value 0, not 100. Place value of 0 is always 0 regardless of position.

• **Sign errors with integers** → When subtracting a negative number, it becomes addition: a - (-b) = a + b. Double negatives yield positive.

• **Missing factor pairs** → When listing factors, students often miss middle pairs. Always work systematically from 1 upward until pairs start repeating.

Quick Reference

• Natural numbers: 1, 2, 3, ... (no zero)
• Whole numbers: 0, 1, 2, 3, ... (zero included)
• Integers: ..., -2, -1, 0, 1, 2, ... (negatives included)
• Place value = digit × value of its position
• 1 is neither prime nor composite; 2 is the smallest and only even prime
• Factors are finite; multiples are infinite
• Divisibility by 6 = divisible by both 2 AND 3