Number System
Overview
The number system forms the bedrock of the Mathematics section in GTET. Questions from this topic test your understanding of fundamental concepts—whole numbers, integers, place value, divisibility rules, HCF and LCM—that appear directly and also underpin arithmetic, algebra and data handling.
Mastery here is non-negotiable. These concepts recur in percentage, ratio, fractions and word problems. A student who deeply understands divisibility and HCF-LCM shortcuts gains speed across the entire quantitative section. Focus on building a solid mental model of number relationships rather than rote memorisation.
Key Concepts
- Natural numbers start from 1 and go infinitely (1, 2, 3, ...). Whole numbers include 0 along with natural numbers (0, 1, 2, 3, ...).
- Integers extend whole numbers to include negatives (..., -3, -2, -1, 0, 1, 2, 3, ...). They form a complete number line with no gaps.
- Place value tells us the value of a digit based on its position. In 5,347: 5 has place value 5,000; 3 has place value 300; 4 has place value 40; 7 has place value 7.
- Face value is the digit itself, regardless of position. Face value of 5 in 5,347 is simply 5.
- Divisibility means one number divides another completely with zero remainder. Divisibility rules provide quick mental tests.
- Factors are numbers that divide a given number exactly. Multiples are products of that number with natural numbers.
- HCF (Highest Common Factor) is the largest number that divides two or more numbers. Also called GCD (Greatest Common Divisor).
- LCM (Lowest Common Multiple) is the smallest number that is a multiple of two or more numbers.
Formulas / Key Facts
Relationship between HCF and LCM: HCF × LCM = Product of the two numbers (For two numbers a and b: HCF(a,b) × LCM(a,b) = a × b)
Divisibility Rules:
| Divisor | Rule |
|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) |
| 3 | Sum of digits divisible by 3 |
| 4 | Last two digits divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 8 | Last three digits divisible by 8 |
| 9 | Sum of digits divisible by 9 |
| 10 | Last digit is 0 |
| 11 | Difference of sums of alternate digits is 0 or divisible by 11 |
Finding HCF — Prime Factorisation Method: Write both numbers as products of primes → Take common primes with lowest powers → Multiply them.
Finding LCM — Prime Factorisation Method: Write both numbers as products of primes → Take all primes with highest powers → Multiply them.
Division Method for HCF: Divide larger by smaller → Divide divisor by remainder → Repeat until remainder is 0 → Last divisor is HCF.
Co-prime numbers: Two numbers with HCF = 1 (e.g., 8 and 15).
Worked Examples
Example 1: Place Value Problem In the number 7,03,829, find the difference between the place value and face value of 3.
Solution:
- Position of 3: Thousands place
- Place value of 3 = 3 × 1,000 = 3,000
- Face value of 3 = 3
- Difference = 3,000 – 3 = 2,997
Example 2: HCF and LCM Calculation Find HCF and LCM of 36 and 48.
Solution: Prime factorisation:
- 36 = 2² × 3²
- 48 = 2⁴ × 3¹
HCF = Take common primes with lowest powers
- Common primes: 2 and 3
- HCF = 2² × 3¹ = 4 × 3 = 12
LCM = Take all primes with highest powers
- LCM = 2⁴ × 3² = 16 × 9 = 144
Verification: HCF × LCM = 12 × 144 = 1,728 = 36 × 48 ✓
Example 3: Word Problem on LCM Three bells toll at intervals of 9, 12 and 15 minutes. If they toll together at 8:00 AM, when will they toll together again?
Solution: Find LCM of 9, 12 and 15:
- 9 = 3²
- 12 = 2² × 3
- 15 = 3 × 5
LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180 minutes = 3 hours
They toll together again at 11:00 AM
Example 4: Divisibility Test Is 2,574 divisible by 6?
Solution: For divisibility by 6, check both 2 and 3:
- Divisibility by 2: Last digit is 4 (even) → Yes
- Divisibility by 3: Sum of digits = 2 + 5 + 7 + 4 = 18, which is divisible by 3 → Yes
Since divisible by both 2 and 3, 2,574 is divisible by 6.
Common Mistakes
- Confusing place value with face value → Place value depends on position (3 in hundreds place = 300); face value is the digit itself (always 3).
- Forgetting that HCF uses lowest powers, LCM uses highest powers → Remember: HCF is the "common minimum" (smaller), LCM is the "combined maximum" (larger).
- Applying the HCF × LCM formula to more than two numbers directly → The formula HCF × LCM = a × b works only for two numbers. For three or more, use prime factorisation.
- Misremembering divisibility rule for 4 vs 8 → For 4, check last two digits. For 8, check last three digits.
- Assuming 1 is prime → 1 is neither prime nor composite. The smallest prime is 2.
- Errors in the division method for HCF → Always divide larger by smaller, then divisor by remainder. Continue until remainder becomes zero—the last divisor (not quotient) is HCF.
Quick Reference
- Whole numbers = 0, 1, 2, 3, ... (natural numbers + zero)
- Integers = ..., -2, -1, 0, 1, 2, ... (include negatives)
- HCF × LCM = Product of two numbers
- Divisibility by 6 = Check both 2 and 3
- For 11: Alternate digit difference = 0 or multiple of 11
- HCF finds largest common divisor; LCM finds smallest common multiple