The Number System forms the foundation of all arithmetic and quantitative reasoning in TS TET Paper I and II. This topic tests your understanding of how numbers are classified, represented, and manipulated—skills essential for teaching primary mathematics effectively.
Expect 3–5 direct questions on whole numbers, integers, place value, divisibility rules, and HCF/LCM calculations. Beyond direct questions, a strong grasp of number properties helps you solve problems across fractions, algebra, and word problems faster. Mastery here means understanding not just procedures but *why* they work—crucial for explaining concepts to young learners.
Focus areas: quick application of divisibility tests, efficient HCF/LCM computation, and confident handling of negative integers.
Key Concepts
**Natural Numbers (N)**: Counting numbers starting from 1. N = {1, 2, 3, 4, ...}. Zero is *not* included.
**Whole Numbers (W)**: Natural numbers plus zero. W = {0, 1, 2, 3, ...}. Every natural number is a whole number.
**Integers (Z)**: Whole numbers and their negatives. Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}. Includes positive integers, negative integers, and zero.
**Place Value vs Face Value**: In 4,725—the face value of 7 is always 7; its place value is 700 (7 × 100). Place value depends on position; face value does not.
**Divisibility**: A number *a* is divisible by *b* if *a* ÷ *b* leaves remainder zero. Divisibility rules provide shortcuts without actual division.
**Factors and Multiples**: Factors divide a number exactly; multiples are products of a number with natural numbers. Example: Factors of 12 are {1, 2, 3, 4, 6, 12}; multiples of 12 are {12, 24, 36, ...}.
**HCF (Highest Common Factor)**: The largest number that divides two or more numbers exactly. Also called GCD.
**LCM (Least Common Multiple)**: The smallest number that is a multiple of two or more numbers.
Formulas / Key Facts
| Concept | Rule / Formula | |---------|----------------| | Divisibility by 2 | Last digit is 0, 2, 4, 6, or 8 | | Divisibility by 3 | Sum of digits is divisible by 3 | | Divisibility by 4 | Last two digits form a number divisible by 4 | | Divisibility by 5 | Last digit is 0 or 5 | | Divisibility by 6 | Divisible by both 2 and 3 | | Divisibility by 8 | Last three digits form a number divisible by 8 | | Divisibility by 9 | Sum of digits is divisible by 9 | | Divisibility by 11 | Difference between sum of odd-place digits and sum of even-place digits is 0 or divisible by 11 | | HCF × LCM | HCF(a, b) × LCM(a, b) = a × b (valid for two numbers) | | Number of factors | If n = p^a × q^b × r^c, then total factors = (a+1)(b+1)(c+1) |
**Integer operations:**
Positive × Positive = Positive
Negative × Negative = Positive
Positive × Negative = Negative
Same rules apply for division
Worked Examples
**Example 1: Place Value** *Find the difference between place value and face value of 6 in 36,472.*
Step 1: Position of 6 → thousands place Step 2: Place value = 6 × 1000 = 6,000 Step 3: Face value = 6 Step 4: Difference = 6,000 – 6 = **5,994**
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**Example 2: Divisibility Test** *Is 2,574 divisible by 6?*
Step 1: Check divisibility by 2 → Last digit is 4 (even). ✓ Step 2: Check divisibility by 3 → Sum of digits = 2 + 5 + 7 + 4 = 18. 18 ÷ 3 = 6. ✓ Step 3: Since divisible by both 2 and 3 → **Yes, divisible by 6**
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**Example 3: HCF and LCM** *Find HCF and LCM of 12 and 18.*
**Prime factorisation method:**
12 = 2² × 3
18 = 2 × 3²
HCF = Product of lowest powers of common primes = 2¹ × 3¹ = **6** LCM = Product of highest powers of all primes = 2² × 3² = 4 × 9 = **36**
**Confusing place value with face value** → Remember: place value changes with position (hundreds, thousands); face value is the digit itself, always constant.
**Forgetting zero is a whole number but not natural** → When counting "natural numbers less than 5," don't include 0. When counting "whole numbers less than 5," include 0.
**Applying the HCF × LCM formula to three numbers** → The formula a × b = HCF × LCM works only for *two* numbers. For three or more, use prime factorisation directly.
**Misapplying divisibility by 4 and 8** → For 4, check last *two* digits; for 8, check last *three* digits. Students often check only the last digit.
**Sign errors with integers** → Subtracting a negative is *adding*: a – (–b) = a + b. Multiplying two negatives gives a *positive*.
**Incorrect sum of digits for divisibility by 3 or 9** → Students sometimes skip digits or add incorrectly. Write digits separately, then add carefully.
Quick Reference
1. **Natural ⊂ Whole ⊂ Integers** — every natural number is whole; every whole number is an integer.
2. **Divisibility by 6** = divisibility by 2 AND 3 (check both).
3. **HCF × LCM = product of the two numbers** — use this for quick verification or finding one when the other is known.
4. **Sum of digits divisible by 9 → number divisible by 9** (also works for 3).
5. **Subtracting a negative → add the positive**: –5 – (–3) = –5 + 3 = –2.
6. **Place value = Face value × Position value** (e.g., 7 in hundreds place → 7 × 100 = 700).
You read the notes — now try one
What is the HCF of 48 and 72?
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