Number System

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

The Number System is the foundation of all quantitative reasoning. HAT-UG tests your ability to work fluently with different types of numbers, perform operations, and apply divisibility rules.

Types of Numbers You Must Know:

Category	Definition	Examples
Natural numbers ($\mathbb{N}$)	Counting numbers starting from 1	1, 2, 3, 4, …
Whole numbers	Natural numbers + 0	0, 1, 2, 3, …
Integers ($\mathbb{Z}$)	Positive, negative, and zero	…, −3, −2, −1, 0, 1, 2, 3, …
Rational numbers ($\mathbb{Q}$)	Ratio of two integers (denominator ≠ 0)	$\frac{3}{4}$, $-\frac{5}{2}$, 0.75, 0.333…
Irrational numbers	Cannot be expressed as a ratio of integers	$\sqrt{2}$, $\pi$, $e$
Real numbers ($\mathbb{R}$)	All rational + irrational numbers	Every number on the number line

Key facts:

• Every integer is a rational number (e.g., $5 = \frac{5}{1}$)
• $\sqrt{n}$ is irrational for any non-perfect square $n$
• Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29…
• 1 is not a prime number — it is neither prime nor composite
• 2 is the only even prime number

Divisibility Rules — Essential for HAT-UG:

Divisor	Rule
2	Last digit is even (0, 2, 4, 6, 8)
3	Sum of digits is divisible by 3
4	Last two digits form a number divisible by 4
5	Last digit is 0 or 5
6	Divisible by both 2 and 3
7	Double the last digit, subtract from the rest — result must be divisible by 7
8	Last three digits form a number divisible by 8
9	Sum of digits is divisible by 9
10	Ends in 0
11	(Sum of digits in odd positions) − (Sum of digits in even positions) is divisible by 11

⚡ HAT-UG Exam Tip: For the 7 rule: Take the last digit, double it (multiply by 2), subtract from the truncated number. Example: 203 → 20 − (3×2) = 20 − 6 = 14 → 14 is divisible by 7, so 203 is divisible by 7. Check: 203 ÷ 7 = 29 ✓.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding and consistent scores.

HCF and LCM — The Two Operations That Appear in Nearly Every HAT-UG Paper:

Prime Factorisation Method:

• HCF = product of the lowest powers of common prime factors
• LCM = product of the highest powers of all prime factors

Example: Find HCF and LCM of 48 and 72.

• $48 = 2^4 \times 3^1$
• $72 = 2^3 \times 3^2$
• HCF = $2^3 \times 3^1 = 8 \times 3 = 24$
• LCM = $2^4 \times 3^2 = 16 \times 9 = 144$

The HCF × LCM Relationship: For any two positive integers $a$ and $b$: $$\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$$ This is extremely useful when you know HCF and LCM and need to find the numbers.

Fraction Operations — Common Pitfalls:

• Adding fractions: find LCD (Least Common Denominator) first
  • $\frac{1}{6} + \frac{3}{8} = \frac{4}{24} + \frac{9}{24} = \frac{13}{24}$
• Multiplying fractions: multiply numerators, multiply denominators, simplify
  • $\frac{3}{4} \times \frac{5}{7} = \frac{15}{28}$
• Dividing fractions: multiply by the reciprocal of the divisor
  • $\frac{3}{4} \div \frac{5}{7} = \frac{3}{4} \times \frac{7}{5} = \frac{21}{20} = 1\frac{1}{20}$

Decimal to Fraction Conversion:

• Terminating decimals: count decimal places → put over 10ⁿ → simplify
  • $0.375 = \frac{375}{1000} = \frac{3}{8}$
• Recurring decimals: use algebra
  • Let $x = 0.\overline{3}$ → $10x = 3.\overline{3}$ → $10x - x = 3$ → $9x = 3$ → $x = \frac{1}{3}$

⚡ Standard Study Tip: Practise the recurring decimal-to-fraction conversion — it appears almost every year in HAT-UG quantitative sections. The shortcut: write $x = 0.\overline{ab}$ → $100x = ab.\overline{ab}$ → $100x - x = ab$ → $99x = ab$ → $x = \frac{ab}{99}$.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory with derivations, historical context, and exam patterns.

Number Theory — The Architecture Underneath the Number System

The Euclidean Algorithm for Finding HCF: The Euclidean algorithm is a systematic method (attributed to Euclid of Alexandria, ~300 BCE) for finding the HCF of two numbers without prime factorisation, which becomes essential for very large numbers.

Steps:

1. Divide the larger number by the smaller number and note the remainder
2. Divide the smaller number by this remainder
3. Repeat until the remainder is 0
4. The last non-zero remainder is the HCF

Example: Find HCF of 252 and 105.

• $252 \div 105 = 2$ remainder $42$ → $252 = 105 \times 2 + 42$
• $105 \div 42 = 2$ remainder $21$ → $105 = 42 \times 2 + 21$
• $42 \div 21 = 2$ remainder $0$ → HCF = 21

Proof that HCF × LCM = a × b: If $d = \text{HCF}(a, b)$, write $a = dm$ and $b = dn$ where $\text{HCF}(m, n) = 1$. Then $\text{LCM}(a, b) = dmn$. Therefore: $d \times \text{LCM} = d \times dmn = dm \times dn = a \times b$. ✓

Base-n Number Systems: HAT-UG sometimes tests conversion between bases. The key formula: $$N = a_k b^k + a_{k-1} b^{k-1} + \dots + a_1 b + a_0$$ where $b$ is the base and $a_i$ are the digits.

Example: Convert $1101_2$ to base 10. $$1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 8 + 4 + 0 + 1 = 13_{10}$$

Modular Arithmetic — The Foundation of Cryptography: When we say $a \equiv b \pmod{n}$, we mean that $a$ and $b$ leave the same remainder when divided by $n$.

• $17 \equiv 2 \pmod{5}$ because $17 \div 5 = 3$ remainder $2$
• Useful property: $(a + b) \mod n = [(a \mod n) + (b \mod n)] \mod n$

HAT-UG Number System — Past Year Patterns (2019–2024):

• 2–3 questions per paper on divisibility rules
• 1–2 questions on HCF/LCM with word problems
• 1 question on decimal-fraction conversion
• 1 question on number system classification (rational/irrational)
• Base conversion appears in approximately 1 in 3 papers

⚡ HAT-UG Advanced Strategy: The fastest way to find LCM of fractions is: $$\text{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{HCF}(b, d)}$$ And for HCF of fractions: $$\text{HCF}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{HCF}(a, c)}{\text{LCM}(b, d)}$$ Commit these to memory — they save enormous time.

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