The Number System is the foundation that every single Quant question at SSC MTS sits on. You cannot do percentages, ratios, or simplification cleanly if this foundation is shaky — so let's nail it once, thoroughly.
Think of numbers the way you think about a family tree. At the very top is the broadest category: Real Numbers (संख्या). Real numbers split into two branches — Rational and Irrational. For SSC MTS, almost everything you deal with is rational.
Rational numbers then branch into Integers (पूर्णांक) — numbers without any fractional part, positive, negative, or zero. Integers branch further into Whole Numbers (0, 1, 2, 3, …) and then Natural Numbers (1, 2, 3, …). Think of it this way: Natural numbers are what you use to count apples at a sabzi mandi. Add zero and you get Whole numbers. Add negatives and you get Integers.
Within Natural numbers, there's a crucial split: Prime vs Composite.
Here's a quick analogy: think of prime numbers as atoms — they cannot be broken down further by multiplication. Composite numbers are molecules — built from two or more atoms (primes).
Beyond classification, the Number System also covers how numbers are structured positionally (place value), how they relate to each other through common divisors and multiples (HCF and LCM), and how division behaves (remainders and quotients).
If you can classify a number, find its factors, compute HCF/LCM quickly, and handle place value without hesitation — you have covered the core of what SSC MTS will test you on in this chapter.
| Category | Definition | Examples | |---|---|---| | Natural Numbers (N) | Positive counting numbers | 1, 2, 3, 4, … | | Whole Numbers (W) | Natural numbers + 0 | 0, 1, 2, 3, … | | Integers (Z) | Whole numbers + negatives | …, -2, -1, 0, 1, 2, … | | Rational Numbers (Q) | Numbers expressible as p/q, q ≠ 0 | 3/4, 0.75, -2, 5 | | Irrational Numbers | Cannot be expressed as p/q | √2, π | | Real Numbers (R) | Rational + Irrational | All of the above |
Relationship: N ⊂ W ⊂ Z ⊂ Q ⊂ R
This distinction appears directly in SSC MTS questions. Do not confuse them.
In 4,736:
4 × 1000 = 40007 × 100 = 7003 × 10 = 306 × 1 = 6Position order from right: Units (10⁰), Tens (10¹), Hundreds (10²), Thousands (10³).
To check if a number n is prime, test divisibility only by primes up to √n.
Primes up to 50 (memorize these): 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. That's 15 primes.
Given any number n, express it as a prime factorization: n = p₁ᵃ × p₂ᵇ × p₃ᶜ …
Total number of factors = (a+1)(b+1)(c+1)…
Example: 12 = 2² × 3¹
Factors = (2+1)(1+1) = 3 × 2 = 6
List them: 1, 2, 3, 4, 6, 12. Confirmed.
HCF (Highest Common Factor / GCD): The largest number that divides both numbers exactly.
Method — Prime Factorization: Express both numbers as prime products. Take the lowest power of each common prime.
12 = 2² × 3¹
18 = 2¹ × 3²
Common primes: 2 and 3. HCF = 2¹ × 3¹ = 6.
LCM (Lowest Common Multiple): The smallest number divisible by both.
Method — Prime Factorization: Take the highest power of every prime that appears.
4 = 2²
6 = 2¹ × 3¹
LCM = 2² × 3¹ = 12.
Golden relationship: HCF × LCM = Product of the two numbers
So for 12 and 18: 6 × LCM = 12 × 18 = 216 → LCM = 36. Verify: 36 is divisible by both 12 and 18. Correct.
When you divide a (dividend) by b (divisor):
a = b × q + r where 0 ≤ r < b
Example: 17 = 5 × 3 + 2. Quotient = 3, Remainder = 2.
SSC MTS often wraps this in word problems: "A number when divided by 5 leaves remainder 2. Which of these could be the number?" — look for numbers of the form 5k + 2: 2, 7, 12, 17, 22…
0.75 = 75/100. Now simplify by dividing numerator and denominator by their HCF.
HCF(75, 100): 75 = 3 × 5², 100 = 2² × 5². HCF = 5² = 25.
75 ÷ 25 = 3, 100 ÷ 25 = 4. So 0.75 = 3/4.
Rule of thumb: for terminating decimals with 2 decimal places, the denominator before simplification is always 100.
12² = 144 is one of 30 squares you should have memorized cold (1² through 30²). But if you blank out:
12² = (10 + 2)² = 100 + 40 + 4 = 144
This identity (a + b)² = a² + 2ab + b² is your fallback for any two-digit square.
When asked "how many factors does N have?" — never list them one by one. Instead, prime-factorize and use the formula.
N = p₁ᵃ × p₂ᵇ × ... → Factors = (a+1)(b+1)...
Example: How many factors does 36 have? 36 = 2² × 3² → (2+1)(2+1) = 9 factors. Standard method (listing): 1,2,3,4,6,9,12,18,36 — takes about 30 seconds to list and verify. Shortcut: 5 seconds to factorize and multiply. That's a 6x speed gain on factor-count questions.
When the question gives you HCF and one number, asks for LCM (or vice versa), never do full prime factorization.
Use: LCM = (Product of numbers) ÷ HCF
Example: Two numbers are 12 and 18. HCF = 6. LCM = ? LCM = (12 × 18) ÷ 6 = 216 ÷ 6 = 36.
Standard method (full LCM by prime factorization): ~40 seconds. This substitution: ~10 seconds. Saves 3 steps entirely.
To square any number near a round number, use (a+b)² = a² + 2ab + b².
Pick the nearest multiple of 10 as 'a', the difference as 'b'.
13² = (10+3)² = 100 + 60 + 9 = 169. 17² = (20-3)² = 400 - 120 + 9 = 289. 15² = (10+5)² = 100 + 100 + 25 = 225.
Standard method (long multiplication): ~25 seconds for a two-digit square. This expansion: ~8 seconds once you're familiar. 3x faster, zero chance of carry errors.
To check if N is prime, test divisibility only by primes ≤ √N. The moment one prime divides N, stop — it's composite.
Is 97 prime? √97 ≈ 9.8. Test: 2 (no — 97 is odd), 3 (9+7=16, not divisible by 3), 5 (doesn't end in 0 or 5), 7 (97 ÷ 7 = 13.8, no). Stop. 97 is prime.
Full trial division up to 97: ~15 checks. Square-root cutoff: 4 checks. That's roughly 4x fewer divisions.
For remainder questions with small divisors, multiply the divisor by the nearest whole quotient mentally and subtract.
What is the remainder when 47 is divided by 6? Nearest multiple of 6 ≤ 47 → 6 × 7 = 42. Remainder = 47 - 42 = 5.
Standard long division written out: ~20 seconds. Mental nearest-multiple subtraction: ~5 seconds. Works cleanly for any divisor ≤ 12.
When you see a Number System question in the exam hall, run through this decision tree in under 5 seconds:
Step 1 — What type of question is it?
a = bq + r, find q by dividing mentally, subtract.Step 2 — Check for traps:
Step 3 — Verify with the option that looks right. Plug back in — takes 5 seconds and saves negative marking.
Why this question: Tests place value identification — one of the most direct Number System questions SSC MTS asks. The trap is confusing face value (7) with place value (700).
Solving path: In 4,736 — read the digits right to left: 6 (units), 3 (tens), 7 (hundreds), 4 (thousands). The digit 7 sits in the hundreds place. Place value = 7 × 100 = 700. Option C.
Why this question: Decimal-to-fraction conversion with simplification — tests whether you know HCF-based simplification, not just writing 75/100 and leaving it there.
Solving path: 0.75 = 75/100. HCF(75,100) = 25. Divide both: 75/25 = 3, 100/25 = 4. Answer = 3/4. Option A. Note: 15/20 is still equal to 3/4 but is not in simplest form — the question says "simplest form", so 3/4 is the only correct answer.
Why this question: Classic definitional trap — tests whether you know 1 is neither prime nor composite, and that 4 is the smallest composite.
Solving path: Check each option. 2 → factors are 1, 2 (exactly two → prime). 3 → factors are 1, 3 (exactly two → prime). 5 → factors are 1, 5 (exactly two → prime). 4 → factors are 1, 2, 4 (three factors → composite). Smallest composite = 4. Option C.
Why this question: HCF by prime factorization — the most foundational HCF question type at this level.
Solving path: 12 = 2² × 3. 18 = 2 × 3². Common primes: 2 and 3. Take lowest powers: 2¹ × 3¹ = 6. Option B. Quick sanity check: 6 divides 12 (yes, 12/6=2) and divides 18 (yes, 18/6=3). Confirmed.
Why this question: Tests the factor-counting formula — a question that trips up students who list factors manually under time pressure.
Solving path: 12 = 2² × 3¹. Number of factors = (2+1)(1+1) = 3 × 2 = 6. Option C. The six factors are 1, 2, 3, 4, 6, 12 — you can verify this takes longer than using the formula.
Treating 1 as prime. The definition of a prime number requires exactly two distinct factors. 1 has only one factor (itself), so it is neither prime nor composite. This is a guaranteed trap in MCQ options.
Confusing HCF with LCM on word problems. "Largest group size" → HCF. "Smallest common interval" (like when two buses depart together again) → LCM. Mixing these up is the single most common Number System error.
Stopping at the unsimplified fraction. When converting a decimal to a fraction, 75/100 and 15/20 are both equal to 3/4 but neither is in simplest form. SSC MTS questions specifically ask for "simplest form" — always divide by the HCF of numerator and denominator.
Using place value = face value. In a number like 5,308, the face value of 3 is 3. The place value of 3 is 300. Do not write 3 when the question asks for place value — re-read the question stem every time.
Not verifying HCF answers. After finding HCF, always do a 3-second check: does the HCF divide both numbers exactly? Students often compute HCF(12,18) = 3 by picking the smallest common prime, forgetting to multiply all common primes together.
Listing factors for large numbers instead of using the formula. For any number with obvious prime factorization, the formula (a+1)(b+1)… is faster and error-free. Listing factors for something like 48 or 60 under exam pressure invites missed entries.