Number System — Study Notes for SSC MTS

Overview

The Number System is the foundation of all quantitative topics in SSC MTS Paper 1. Questions appear directly (classify number types, apply divisibility rules, work with fractions) and indirectly (every calculation question uses these concepts). Expect 3–5 direct questions from this topic and many more that require quick mental math based on number properties.

Mastery here means speed: identifying odd/even patterns instantly, applying divisibility tests without writing steps, and converting fractions-decimals fluently. The exam tests both conceptual understanding (is √2 rational?) and computational skill (simplify 3/7 + 2/5 in 15 seconds). Students who internalize divisibility rules and fraction shortcuts gain 2–3 minutes per paper — a decisive advantage in a time-bound exam.

Focus on integers, fractions (proper, improper, mixed), decimals, rational/irrational classification, and the big six divisibility rules (2, 3, 5, 9, 10, 11). Skip advanced number theory; SSC MTS stays with practical, calculation-heavy problems.

Key Concepts

• **Natural numbers** (1, 2, 3, ...) count discrete objects. **Whole numbers** add zero (0, 1, 2, ...). **Integers** include negatives (..., -2, -1, 0, 1, 2, ...). Every natural number is whole and every whole number is an integer.

• **Rational numbers** can be written as p/q where p and q are integers and q ≠ 0. All terminating and repeating decimals are rational (e.g. 0.75 = 3/4, 0.333... = 1/3). **Irrational numbers** cannot be expressed as fractions (e.g. √2, π, √3).

• **Even numbers** are divisible by 2 (last digit 0, 2, 4, 6, 8). **Odd numbers** are not (last digit 1, 3, 5, 7, 9). Sum of two evens is even; sum of two odds is even; sum of even and odd is odd.

• **Prime numbers** have exactly two factors: 1 and the number itself (2, 3, 5, 7, 11, 13, ...). Note: 1 is neither prime nor composite. 2 is the only even prime.

• **Composite numbers** have more than two factors (4, 6, 8, 9, 10, ...). Every composite number can be expressed as a product of primes (prime factorization).

• **Fractions**: Proper fraction (numerator < denominator, e.g. 3/4), improper fraction (numerator ≥ denominator, e.g. 7/4), mixed number (whole + proper fraction, e.g. 1¾). Convert improper to mixed: divide numerator by denominator; quotient is whole part, remainder over denominator is fractional part.

• **Divisibility rules** let you test if one number divides another without performing division. Essential for simplification, factorization, and LCM/HCF problems.

• **Place value** vs **face value**: In 3,425, the digit 4 has face value 4 but place value 400 (hundreds place). Sum of digits = 3+4+2+5 = 14.

Formulas / Key Facts

• **Divisibility by 2**: Last digit is 0, 2, 4, 6 or 8. Example: 134 is divisible by 2; 137 is not.

• **Divisibility by 3**: Sum of all digits is divisible by 3. Example: 123 → 1+2+3 = 6 → divisible by 3.

• **Divisibility by 5**: Last digit is 0 or 5. Example: 125 and 340 are divisible by 5.

• **Divisibility by 9**: Sum of all digits is divisible by 9. Example: 729 → 7+2+9 = 18 → divisible by 9.

• **Divisibility by 10**: Last digit is 0. Example: 230 is divisible by 10.

• **Divisibility by 11**: Difference between sum of digits at odd places and sum of digits at even places is 0 or divisible by 11. Example: 1331 → (1+3) - (3+1) = 0 → divisible by 11.

• **Fraction addition/subtraction with different denominators**: Find LCM of denominators, convert each fraction, then add/subtract numerators. Example: 2/3 + 3/4 = 8/12 + 9/12 = 17/12.

• **Fraction multiplication**: Multiply numerators, multiply denominators. Example: 2/3 × 4/5 = 8/15.

• **Fraction division**: Multiply by the reciprocal. Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.

• **Decimal to fraction**: Write as fraction over power of 10, then simplify. Example: 0.75 = 75/100 = 3/4.

• **Recurring decimal to fraction**: For 0.333..., let x = 0.333..., then 10x = 3.333..., so 10x - x = 3, thus x = 3/9 = 1/3.

Worked Examples

**Example 1**: Is 5628 divisible by both 3 and 11?

*Solution*:

• Test for 3: Sum of digits = 5+6+2+8 = 21. Since 21 is divisible by 3, 5628 is divisible by 3.
• Test for 11: Odd places sum = 5+2 = 7; Even places sum = 6+8 = 14. Difference = 14-7 = 7. Since 7 is not 0 or divisible by 11, 5628 is not divisible by 11.
• Answer: Divisible by 3, not by 11.

**Example 2**: Simplify 5/6 + 7/9.

*Solution*:

• LCM of 6 and 9 is 18.
• Convert: 5/6 = (5×3)/(6×3) = 15/18; 7/9 = (7×2)/(9×2) = 14/18.
• Add: 15/18 + 14/18 = 29/18.
• Answer: 29/18 or 1 11/18.

**Example 3**: Convert the mixed number 3 2/5 to an improper fraction.

*Solution*:

• Multiply whole part by denominator: 3×5 = 15.
• Add numerator: 15+2 = 17.
• Write over original denominator: 17/5.
• Answer: 17/5.

**Example 4**: Which of the following is irrational? (a) 0.75 (b) 2/3 (c) √5 (d) 0.666...

*Solution*:

• (a) 0.75 = 3/4, rational.
• (b) 2/3 is already a fraction, rational.
• (c) √5 cannot be expressed as p/q, irrational.
• (d) 0.666... = 2/3, rational.
• Answer: (c) √5.

Common Mistakes

• **Confusing natural and whole numbers**: Students forget that 0 is whole but not natural. Fix: Natural starts at 1; whole starts at 0.

• **Wrong divisibility by 11**: Applying the alternating sum rule incorrectly (forgetting which place is odd/even or miscalculating). Fix: Start from the rightmost digit as position 1 (odd), consistently alternate, and remember to check if difference is 0 or multiple of 11.

• **Adding fractions without common denominator**: Writing 2/3 + 1/4 = 3/7 (adding tops and bottoms separately). Fix: Always find LCM of denominators first, convert, then add only numerators.

• **Treating √4 and √2 the same way**: √4 = 2 is rational (perfect square), but √2 is irrational. Fix: Only square roots of perfect squares are rational.

• **Misclassifying 1 as prime**: 1 has only one factor (itself), so it is neither prime nor composite. Fix: Prime requires exactly two distinct factors.

Quick Reference

• Divisibility shortcuts: 2 (last digit even), 3 (digit sum ÷ 3), 5 (last digit 0 or 5), 9 (digit sum ÷ 9), 10 (last digit 0), 11 (alternating digit sum difference ÷ 11).
• Rational = can write as fraction; irrational = cannot (like √2, π).
• Even ± even = even; odd ± odd = even; even ± odd = odd.
• Improper to mixed: divide numerator by denominator, quotient = whole, remainder/denominator = fraction.
• Fraction ops: add/subtract (common denominator), multiply (straight across), divide (multiply by reciprocal).
• 1 is neither prime nor composite; 2 is the only even prime.