Number System — SSC CHSL Study Notes
Overview
The Number System forms the backbone of Quantitative Aptitude in SSC CHSL. Expect 3–5 direct questions from this topic in Tier 1, plus countless indirect applications across other arithmetic topics. Mastery here speeds up your entire math section.
This topic tests your understanding of how numbers behave: divisibility patterns, prime factorization, finding HCF and LCM, working with fractions and decimals, and simplifying surds. SSC loves testing these concepts through word problems, formula applications, and quick mental math scenarios. Students who internalize divisibility rules and master fraction–decimal conversions save 20–30 seconds per question elsewhere in the paper.
Focus on pattern recognition rather than lengthy calculations. Many CHSL questions reward spotting shortcuts — like recognizing that a number divisible by both 3 and 4 must be divisible by 12, or converting recurring decimals to fractions instantly.
Key Concepts
• **Natural, Whole, and Integers**: Natural numbers (1, 2, 3…), whole numbers (0, 1, 2…), integers (…−2, −1, 0, 1, 2…). Know the difference; questions often test which set a result belongs to.
• **Prime and Composite Numbers**: A prime has exactly two factors (1 and itself); 2 is the only even prime. Composite numbers have more than two factors. 1 is neither prime nor composite.
• **Divisibility Rules**: Quick mental checks — divisibility by 2 (last digit even), by 3 (sum of digits divisible by 3), by 4 (last two digits divisible by 4), by 5 (ends in 0 or 5), by 6 (divisible by both 2 and 3), by 8 (last three digits divisible by 8), by 9 (sum of digits divisible by 9), by 11 (difference of alternate digit sums divisible by 11).
• **HCF and LCM**: HCF (Highest Common Factor) is the largest number dividing all given numbers. LCM (Lowest Common Multiple) is the smallest number all given numbers divide into. For two numbers a and b: **HCF × LCM = a × b**.
• **Fractions**: Proper fraction (numerator < denominator), improper fraction (numerator ≥ denominator), mixed number (whole + proper fraction). Convert between forms fluently.
• **Decimals**: Terminating decimals end (0.25), non-terminating repeating decimals recur (0.333…). Any fraction with denominator having only 2s and 5s as prime factors terminates; otherwise it recurs.
• **Surds**: Irrational roots like √2, √3, ∛5. Rationalization means removing surds from denominators by multiplying by the conjugate (e.g., multiply 1/√2 by √2/√2 to get √2/2).
• **Co-primes**: Two numbers are co-prime if their HCF is 1 (e.g., 8 and 15). Co-primes need not be prime themselves.
Formulas / Key Facts
• **Product formula**: For any two numbers a and b, **a × b = HCF(a,b) × LCM(a,b)**
• **LCM of fractions** = LCM of numerators / HCF of denominators
• **HCF of fractions** = HCF of numerators / LCM of denominators
• **Number of factors** of n = p₁^a × p₂^b × p₃^c is **(a+1)(b+1)(c+1)** where p₁, p₂, p₃ are distinct primes
• **Sum of factors** of n = p^a is **(p^(a+1) − 1)/(p − 1)**
• **Converting recurring decimal to fraction**: For 0.777… = 7/9; for 0.343434… = 34/99; for 0.5232323… = (523−5)/990
• **Rationalizing denominator**: 1/(a+√b) = (a−√b)/(a²−b); 1/√a = √a/a
• **Perfect square property**: A perfect square has an odd number of total factors
• **Unit digit patterns**: Powers of 2 repeat every 4 (2,4,8,6…); powers of 3 repeat every 4 (3,9,7,1…); powers of 4 repeat every 2 (4,6…)
Worked Examples
**Example 1: Find HCF and LCM of 12 and 18**
Prime factorization: 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 = 36
Verify: 12 × 18 = 216 = 6 × 36 ✓
**Example 2: How many factors does 360 have?**
360 = 2³ × 3² × 5¹
Number of factors = (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24 factors
**Example 3: Convert 0.28̄ (28 repeating) to a fraction**
Let x = 0.282828…
Then 100x = 28.282828…
Subtract: 100x − x = 28
99x = 28
x = 28/99
**Example 4: Rationalize 5/(3+√2)**
Multiply numerator and denominator by conjugate (3−√2):
= 5(3−√2) / [(3+√2)(3−√2)]
= 5(3−√2) / (9−2)
= 5(3−√2) / 7
= (15−5√2) / 7
**Example 5: Find the largest number that divides 245 and 1029 leaving remainders 5 in each case**
Required HCF divides (245−5) and (1029−5) exactly
Find HCF of 240 and 1024
240 = 2⁴ × 3 × 5, 1024 = 2¹⁰
HCF = 2⁴ = 16
Common Mistakes
**Mistake 1**: Confusing HCF with LCM in word problems → **Fix**: HCF for "greatest number that divides", "maximum identical groups"; LCM for "smallest common multiple", "bells ringing together again"
**Mistake 2**: Adding fractions without finding common denominator → **Fix**: Always convert to common denominator first: 1/3 + 1/4 = 4/12 + 3/12 = 7/12, not 2/7
**Mistake 3**: Believing 1 is a prime number → **Fix**: By definition, primes have exactly two factors; 1 has only one factor (itself), so it's neither prime nor composite
**Mistake 4**: Incorrectly rationalizing complex surds → **Fix**: Use conjugate properly: for a+√b use a−√b; for √a+√b use √a−√b; multiply both numerator and denominator
**Mistake 5**: Thinking all decimals terminate → **Fix**: Check denominator's prime factors after reducing the fraction; only 2s and 5s give terminating decimals; any other prime factor means recurring decimal
**Mistake 6**: Calculating LCM by multiplying the numbers directly → **Fix**: Use LCM = (a × b)/HCF or prime factorization method; direct multiplication gives wrong answer unless numbers are co-prime
Quick Reference
• **Divisibility by 3**: Sum of digits divisible by 3; by 9: sum divisible by 9; by 11: alternating digit-sum difference divisible by 11
• **HCF × LCM = Product of two numbers** — use this to find one when you know the other three
• **Recurring decimal x̄**: divide by 9; xy̅: divide by 99; xyz̅: divide by 999, then simplify
• **Number of factors formula**: (a+1)(b+1)(c+1)… where exponents are a, b, c… in prime factorization
• **Rationalize 1/√n** by multiplying top and bottom by √n; for 1/(a+√b) multiply by (a−√b)
• **Co-prime check**: Two numbers with HCF = 1; their LCM equals their product