Classification of Numbers
Every number you encounter in mathematics โ from the price of a bus ticket to the digits of ฯ โ belongs to one or more carefully defined families, and knowing these families is the foundation of every quantitative question in SSC CGL and similar competitive exams.
Definition: A number system is a structured classification of numbers into sets based on their properties, such as whether they can be expressed as a ratio, whether they have a repeating decimal expansion, and how they behave under arithmetic operations.
Natural Numbers (N)
Definition: The set of natural numbers is N = {1, 2, 3, 4, 5, โฆ}. These are the counting numbers โ the ones a child first learns to count objects with.
Key properties:
- Smallest natural number: 1
- No upper bound โ they continue infinitely
- Closed under addition and multiplication (adding or multiplying two natural numbers always gives a natural number)
- NOT closed under subtraction (3 โ 5 = โ2, which is not in N)
Why it matters: Every counting operation โ ranking candidates in an exam, counting factories in a district โ involves natural numbers.
Whole Numbers (W)
Definition: W = {0, 1, 2, 3, 4, โฆ} โ natural numbers plus zero.
- Smallest whole number: 0
- Zero is the additive identity: a + 0 = a for any whole number a
- Every natural number is also a whole number, but 0 is not a natural number
Common misconception: Many students treat "natural numbers" and "whole numbers" as synonyms. They are not โ the difference is the presence of zero.
Integers (Z)
Definition: Z = {โฆ, โ3, โ2, โ1, 0, 1, 2, 3, โฆ} โ all positive whole numbers, zero, and all negative whole numbers.
- Integers are closed under addition, subtraction, and multiplication
- Division of two integers need not be an integer (7 รท 2 = 3.5)
- The letter Z comes from the German word Zahlen (numbers)
Real-world example: Temperature in Shimla in January can be โ5ยฐC, โ3ยฐC, or 2ยฐC โ all integers.
Rational Numbers (Q)
Definition: A rational number is any number that can be written in the form p/q, where p and q are integers and q โ 0.
Examples: 3/4, โ7/2, 0.5 (= 1/2), 0.333โฆ (= 1/3), 6 (= 6/1)
Key insight โ decimal behaviour:
- A rational number always has either a terminating decimal (e.g., 1/4 = 0.25) or a recurring (repeating) decimal (e.g., 1/3 = 0.333โฆ)
- If you see a decimal that terminates or repeats, it is rational
Why it matters: Fractions appear in ratio, proportion, percentage, and speed-distance problems โ all high-frequency SSC topics.
Irrational Numbers
Definition: An irrational number cannot be expressed as p/q for any integers p and q. Its decimal expansion is non-terminating and non-repeating.
Common irrational numbers to memorise:
- โ2 โ 1.41421โฆ
- โ3 โ 1.73205โฆ
- โ5 โ 2.23606โฆ
- ฯ โ 3.14159โฆ
- e โ 2.71828โฆ
Common misconception: Students often think 22/7 = ฯ. It does NOT โ 22/7 is a rational approximation; ฯ itself is irrational.
Another trap: โ4 = 2, which IS rational. Not every square root is irrational โ only the square root of a non-perfect-square is irrational.
Real Numbers (R)
Definition: The set of real numbers is the union of all rational and irrational numbers. Every point on the number line corresponds to exactly one real number.
| Set | Notation | Contains | Example |
|---|---|---|---|
| Natural | N | {1, 2, 3, โฆ} | 7 |
| Whole | W | {0, 1, 2, 3, โฆ} | 0 |
| Integer | Z | {โฆโ2,โ1, 0, 1, 2โฆ} | โ5 |
| Rational | Q | p/q, qโ 0 | 3/4 |
| Irrational | โ | Non-repeating decimals | โ2 |
| Real | R | Q โช Irrational | All of the above |
The hierarchy: N โ W โ Z โ Q โ R
Prime Numbers
Definition: A prime number has exactly two distinct positive factors: 1 and itself.
Important facts:
- 2 is the only even prime number
- 1 is NOT prime (it has only one factor: itself)
- The number of primes below 100 is 25
List of all 25 primes below 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Why it matters: Prime factorisation is the engine behind LCM, HCF, number of divisors, and many SSC number theory problems.
Composite Numbers
Definition: A composite number has more than two factors โ i.e., it can be divided by at least one number other than 1 and itself.
Examples: 4 (factors: 1, 2, 4), 12 (factors: 1, 2, 3, 4, 6, 12)
Key fact: Every composite number can be expressed as a unique product of primes (Fundamental Theorem of Arithmetic).
Example: 360 = 2ยณ ร 3ยฒ ร 5
Special case: 1 is neither prime nor composite. This is one of the most frequently tested tricky facts in SSC CGL.
Co-prime Numbers
Definition: Two numbers are co-prime (or relatively prime) if their HCF = 1 โ they share no common factor other than 1.
Examples:
- 8 and 15: factors of 8 = {1, 2, 4, 8}; factors of 15 = {1, 3, 5, 15}; HCF = 1 โ co-prime โ
- 4 and 9: HCF = 1 โ co-prime โ
- 6 and 9: HCF = 3 โ NOT co-prime โ
Key insight: Co-prime numbers need not be prime themselves โ 8 and 15 are co-prime even though neither is prime.
Why it matters: Co-primality appears in LCM/HCF questions and in fraction simplification problems.
Even and Odd Numbers
Even numbers: Divisible by 2. Last digit is 0, 2, 4, 6, or 8.
Odd numbers: Not divisible by 2. Last digit is 1, 3, 5, 7, or 9.
Arithmetic rules (heavily tested in SSC):
- Even + Even = Even
- Odd + Odd = Even
- Even + Odd = Odd
- Even ร Even = Even
- Odd ร Odd = Odd
- Even ร Odd = Even
Perfect Numbers
Definition: A perfect number equals the sum of all its proper divisors (all divisors except itself).
Example: 6 โ proper divisors: 1, 2, 3 โ sum = 6 โ
Example: 28 โ proper divisors: 1, 2, 4, 7, 14 โ sum = 28 โ
The next perfect number after 28 is 496. For SSC purposes, just remember 6 and 28.
Worked Example โ Classifying a Number
Question: Classify the number 0.171717โฆ and determine which sets it belongs to.
Solution:
Step 1: The decimal 0.1717โฆ has a repeating block "17". A non-terminating but repeating decimal is always rational.
Step 2: Express as fraction: Let x = 0.1717โฆ
Then 100x = 17.1717โฆ
100x โ x = 17 โ 99x = 17 โ x = 17/99
Step 3: 17/99 is in the form p/q (p=17, q=99, qโ 0).
Conclusion: 0.1717โฆ is a rational number, and therefore also a real number. It is not a natural number, whole number, or integer.
- โ- Natural numbers start at 1; whole numbers start at 0; integers include negatives
- โ- Every natural โ whole โ integer โ rational โ real
- โ- Rational decimals terminate or repeat; irrational decimals never repeat
- โ- 2 is the only even prime number
- โ- 1 is neither prime nor composite โ this is a favourite SSC trap
- โ- Co-prime means HCF = 1, not that either number is prime
- โ- There are exactly 25 prime numbers less than 100
- โ- Perfect numbers: 6, 28 (sum of proper divisors equals the number)
"Never Will I Quit Reaching" โ Natural, Whole, Integer, Rational, Real โ each set contains the previous one. For primes: "2 is the lone ranger" โ the only even prime.
- โ- Numbers split into Natural โ Whole โ Integer โ Rational โ Real (each is a subset of the next)
- โ- Rational numbers have terminating or recurring decimals; irrational numbers have neither
- โ- Prime: exactly 2 factors; Composite: more than 2; 1: neither
- โ- Co-prime pairs have HCF = 1 (e.g., 8 and 15)
- โ- Even/odd arithmetic rules are directly tested in SSC CGL
- โ- 25 primes exist below 100; memorise the full list
Key Properties of Special Numbers
Certain numbers have such special mathematical properties that they appear as traps and shortcuts in competitive exams again and again โ mastering them means converting hard questions into one-step recalls.
Prime Numbers up to 100
Every serious SSC CGL aspirant must have the complete list of 25 primes below 100 memorised:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97
Why memorise the full list? SSC CGL regularly asks "which of the following is prime?" with options like 91, 97, 87, 93 โ if you know the list, the answer is instant.
Grouping trick for memory:
- Teens primes: 11, 13, 17, 19 (all four โ skip 15)
- Twenties primes: only 23 and 29
- Thirties primes: 31, 37 (only two)
- Forties primes: 41, 43, 47 (three)
- Fifties primes: 53, 59 (only two โ skip 51=3ร17, 55=5ร11, 57=3ร19)
- Sixties primes: 61, 67 (only two โ 63=9ร7, 65=5ร13)
- Seventies primes: 71, 73, 79 (three)
- Eighties primes: 83, 89 (only two โ 81=3โด, 87=3ร29)
- Ninety primes: 97 only (91=7ร13 is a classic trap!)
How to Test if a Number is Prime
Definition: To check if n is prime, test divisibility by all prime numbers up to โn.
Why only up to โn? If n has a factor larger than โn, then its corresponding paired factor must be smaller than โn โ so you would have found it already.
Worked illustration:
- Test n = 221. โ221 โ 14.8. Check primes: 2, 3, 5, 7, 11, 13.
- 221 รท 13 = 17. So 221 = 13 ร 17. Composite.
- Test n = 233. โ233 โ 15.3. Check 2, 3, 5, 7, 11, 13: none divide 233. Prime.
Perfect Numbers
Definition: A perfect number is a positive integer that equals the sum of all its proper divisors (all positive divisors excluding the number itself).
The first two perfect numbers (and the only ones relevant for SSC CGL):
| Number | Proper Divisors | Sum |
|---|---|---|
| 6 | 1, 2, 3 | 6 โ |
| 28 | 1, 2, 4, 7, 14 | 28 โ |
The next perfect number is 496, then 8128 โ far beyond competitive exam scope.
Why it matters: "Which of the following is a perfect number?" is a direct recall question.
Common misconception: Students sometimes confuse "perfect number" with "perfect square." A perfect square is nยฒ for integer n (4, 9, 16, 25โฆ). These are entirely different concepts.
Twin Primes
Definition: Twin primes are pairs of prime numbers that differ by exactly 2.
Known twin prime pairs (up to 100):
- (3, 5)
- (5, 7)
- (11, 13)
- (17, 19)
- (29, 31)
- (41, 43)
- (59, 61)
- (71, 73)
Note: (2, 3) differ by 1, not 2 โ not twin primes.
Twin Prime Conjecture: It is believed (but not proven) that there are infinitely many twin prime pairs. This is one of the great unsolved problems in mathematics.
Even and Odd Number Properties
These arithmetic rules are tested both directly and embedded in larger problems:
| Operation | Result | Example |
|---|---|---|
| Even + Even | Even | 4 + 6 = 10 |
| Odd + Odd | Even | 3 + 7 = 10 |
| Even + Odd | Odd | 4 + 3 = 7 |
| Even ร Even | Even | 4 ร 6 = 24 |
| Odd ร Odd | Odd | 3 ร 7 = 21 |
| Even ร Odd | Even | 4 ร 3 = 12 |
| Even โ Even | Even | 8 โ 4 = 4 |
| Odd โ Odd | Even | 7 โ 3 = 4 |
Key insight for SSC: The product of ANY set of numbers is even if at least one of them is even. The product is odd only if all numbers in the product are odd.
Real-world example: If you buy 3 items priced at โน5, โน7, and โน9 each (all odd), total = โน21 (odd). Add one item at โน4 (even) and the product of prices is now even.
Properties of Powers (Even/Odd Exponents)
For SSC number theory and remainder problems:
- Even number raised to any positive power โ Even
- Odd number raised to any positive power โ Odd
- The last digit of a power follows a cycle (tested in cyclicity problems)
Last digit cycles for common bases:
- 2: cycle 2, 4, 8, 6 (period 4)
- 3: cycle 3, 9, 7, 1 (period 4)
- 7: cycle 7, 9, 3, 1 (period 4)
- 8: cycle 8, 4, 2, 6 (period 4)
- 1, 5, 6: last digit never changes (always 1, 5, 6 respectively)
Composite Number Properties
Every composite number n can be written as a product of primes in exactly one way (up to order) โ this is the Fundamental Theorem of Arithmetic.
Number of divisors formula: If n = pโ^a ร pโ^b ร pโ^c โฆ, then number of divisors = (a+1)(b+1)(c+1)โฆ
Example: 360 = 2ยณ ร 3ยฒ ร 5ยน
Number of divisors = (3+1)(2+1)(1+1) = 4 ร 3 ร 2 = 24
Worked Example โ Applying Multiple Properties
Question: The product of two prime numbers is 77. What is their sum?
Solution:
Step 1: Express 77 as a product. 77 = 7 ร 11.
Step 2: Check: 7 is prime (not divisible by 2, 3, 5; โ7 < 3). 11 is prime.
Step 3: Sum = 7 + 11 = 18.
Question: How many prime numbers are there between 50 and 70?
Solution:
Step 1: List candidates: 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69.
Step 2: Eliminate composites: 51=3ร17, 52=2ร26, 54=2ร27, 55=5ร11, 56=2ร28, 57=3ร19, 58=2ร29, 60=2ร30, 62=2ร31, 63=7ร9, 64=2โถ, 65=5ร13, 66=2ร33, 68=2ร34, 69=3ร23.
Step 3: Remaining primes: 53, 59, 61, 67 โ 4 primes.
- โ- There are 25 primes below 100; 91=7ร13 is a notorious non-prime trap
- โ- To test primality of n, check divisibility by primes up to โn only
- โ- Perfect numbers: 6 and 28 (proper divisors sum to the number)
- โ- Twin primes differ by exactly 2; pairs up to 100: (3,5),(5,7),(11,13),(17,19),(29,31),(41,43),(59,61),(71,73)
- โ- Product of numbers is odd ONLY IF every factor is odd
- โ- Sum of two odd numbers is always even
- โ- Number of divisors of n = p^a ร q^b is (a+1)(b+1)
"Perfect: Six and Twenty-Eight" โ 6 = 1+2+3; 28 = 1+2+4+7+14. For twin primes: think of prime siblings sitting next to each other on the number line with one seat gap.
- โ- Full prime list below 100 has exactly 25 numbers; memorise by decade groups
- โ- Primality test: divide by primes โค โn; if none divide, it is prime
- โ- Perfect number = sum of proper divisors; 6 and 28 are the exam-relevant ones
- โ- Twin primes are prime pairs with difference 2
- โ- Product is even if any factor is even; product is odd only if all factors are odd
- โ- Number of divisors uses the prime factorisation exponent formula (a+1)(b+1)โฆ
Solved Example: Identifying Number Types
The fastest route to a correct answer in SSC CGL is not just knowing the definition of a prime number โ it's being able to identify one (or rule it out) in under 30 seconds without a calculator.
The Core Technique: Square Root Boundary Test
Definition: A number n is prime if and only if it has no prime factor less than or equal to โn.
This works because:
- If n = a ร b and both a > โn and b > โn, then a ร b > n โ contradiction.
- So at least one factor must be โค โn.
- Therefore, if no prime โค โn divides n, then n has no factor at all (other than 1 and n), so it is prime.
Practical rule:
- For numbers up to 100: check divisibility by 2, 3, 5, 7 (since โ100 = 10, and 7 is the largest prime below 10)
- For numbers up to 200: also check 11 and 13 (since โ200 โ 14.1)
- For numbers up to 400: also check 17 and 19 (since โ400 = 20)
Divisibility Quick-Checks
Before doing division, use divisibility rules to eliminate candidates instantly:
| Divisor | Rule | Example |
|---|---|---|
| 2 | Last digit even | 84: last digit 4 โ divisible |
| 3 | Sum of digits divisible by 3 | 93: 9+3=12 โ divisible |
| 5 | Last digit 0 or 5 | 95: last digit 5 โ divisible |
| 7 | No simple rule; just divide | |
| 11 | (Sum of odd-position digits) โ (Sum of even-position digits) = 0 or 11 | 121: (1+1)โ2=0 โ divisible |
| 13 | No simple rule; just divide |
Solved Example 1 (from the lesson stub)
Question: Which of the following is a prime number โ 91, 97, 87, 93?
Solution:
Step 1: Examine 91. โ91 โ 9.5 โ check primes 2, 3, 5, 7.
- Is 91 even? No.
- Digit sum = 9+1 = 10 โ not divisible by 3.
- Last digit not 0 or 5 โ not divisible by 5.
- 91 รท 7 = 13. Yes! So 91 = 7 ร 13. Composite.
Step 2: Examine 87. โ87 โ 9.3 โ check 2, 3, 5, 7.
- Digit sum = 8+7 = 15 โ divisible by 3. 87 = 3 ร 29. Composite.
Step 3: Examine 93. โ93 โ 9.6 โ check 2, 3, 5, 7.
- Digit sum = 9+3 = 12 โ divisible by 3. 93 = 3 ร 31. Composite.
Step 4: Examine 97. โ97 โ 9.8 โ check 2, 3, 5, 7.
- Not even. Digit sum = 9+7 = 16 โ not divisible by 3. Last digit โ 0,5. 97 รท 7 = 13.857โฆ Not divisible.
- No prime โค โ97 divides 97. 97 is prime.
Conclusion: 97 is the prime.
Why 91 is the classic trap: 91 looks prime because it is not divisible by 2, 3, or 5. But 7 ร 13 = 91 โ always check 7 for two-digit numbers!
Solved Example 2: Identifying the Type of a Given Number
Question: Classify: 1, 2, 4, 15, 17, 28, 36.
Solution:
| Number | Factors | Classification |
|---|---|---|
| 1 | 1 only | Neither prime nor composite |
| 2 | 1, 2 | Prime (only even prime) |
| 4 | 1, 2, 4 | Composite |
| 15 | 1, 3, 5, 15 | Composite |
| 17 | 1, 17 | Prime |
| 28 | 1, 2, 4, 7, 14, 28 | Composite; also a perfect number |
| 36 | 1,2,3,4,6,9,12,18,36 | Composite; perfect square (6ยฒ) |
Solved Example 3: How Many Primes Between Two Numbers?
Question: How many prime numbers lie between 80 and 100?
Solution:
Step 1: Candidates: 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99.
Step 2: Eliminate composites quickly โ
- Even numbers: 82, 84, 86, 88, 90, 92, 94, 96, 98 โ all composite
- Divisible by 5: 85 (=5ร17), 90, 95 (=5ร19) โ composite
- Divisible by 3: 81 (=3โด), 87 (=3ร29), 93 (=3ร31), 99 (=9ร11) โ composite
- 91 = 7 ร 13 โ composite
Step 3: Remaining: 83 and 89 and 97.
- 83: โ83 โ 9.1. Not divisible by 2, 3, 5, 7. โ Prime
- 89: โ89 โ 9.4. Not divisible by 2, 3, 5, 7. โ Prime
- 97: already confirmed prime above.
Conclusion: 3 primes between 80 and 100 (83, 89, 97).
Common SSC Traps
Trap 1 โ 1 is prime: FALSE. 1 is neither prime nor composite.
Trap 2 โ 91 is prime: FALSE. 91 = 7 ร 13.
Trap 3 โ 2 is not prime (it's even): FALSE. 2 is the only even prime.
Trap 4 โ All odd numbers are prime: FALSE. 9 = 3ร3, 15 = 3ร5, 21 = 3ร7 are all odd composites.
Trap 5 โ โn must be an integer to be relevant: FALSE. Even if โn is not an integer (e.g., โ91 โ 9.5), you still check all primes up to the floor of โn.
Worked Shortcut: Composite Identification by Digit Sum
Question: Without dividing, determine whether 561 is prime.
Solution:
Step 1: Digit sum = 5+6+1 = 12. Divisible by 3 โ 561 is divisible by 3. Composite. (561 = 3 ร 187)
This digit-sum trick eliminates most large numbers before you even attempt division.
- โ- To test primality of n, check only primes โค โn โ once you pass โn, stop
- โ- For numbers 1โ100: checking 2, 3, 5, 7 is sufficient (โ100 = 10)
- โ- 91 = 7 ร 13 is the most common prime-impersonator in SSC exams
- โ- 1 is neither prime nor composite โ a guaranteed trick question
- โ- Use divisibility rules (digit sum for 3; last digit for 2, 5) before attempting division
- โ- Between 80 and 100 there are exactly 3 primes: 83, 89, 97
- โ- Perfect numbers (6, 28) and twin primes are separate special categories
"Seven Traps Ninety-One" โ whenever you see 91, immediately check รท7. For the test: "Check up to the Square Root, not a step more."
- โ- Primality test: find โn, then test divisibility by each prime up to that bound
- โ- Apply divisibility rules first (2, 3, 5) to quickly eliminate most candidates
- โ- 91 = 7ร13, 87 = 3ร29, 93 = 3ร31 โ all favourite composite traps in MCQ options
- โ- 97 is the largest prime below 100
- โ- 1 is not prime; 2 is prime and is the only even prime
- โ- Between any two consecutive perfect squares, count primes carefully using the method above
โก Speed Tricks & Shortcuts
- To test if n is prime, only trial-divide by primes โค โn (e.g. for 149, check up to 12).
- 2 is the only even prime; 1 is neither prime nor composite.
- Two numbers are co-prime if their HCF = 1 (they needn't be prime themselves, e.g. 8 & 9).
- Product of n consecutive integers is always divisible by n!.
Don't call 1 a prime and don't forget 2 โ "smallest prime is 2, smallest odd prime is 3" is a classic trap.
Types of Numbers โ Revision Notes
Quick-revision notes for Types of Numbers โ the must-know points for SSC CGL Tier-I/II.
- Natural numbers (1,2,3โฆ), Whole numbers (0,1,2โฆ), Integers (โฆโ2,โ1,0,1,2โฆ), and their subsets.
- Rational numbers can be written as p/q (qโ 0); irrational numbers (like โ2, ฯ) cannot.
- Prime numbers have exactly two factors (1 and itself); 2 is the only even prime; 1 is neither prime nor composite.
- Composite numbers have more than two factors; co-prime numbers share HCF 1.
- Even numbers are divisible by 2; odd are not. Perfect numbers equal the sum of their proper divisors (e.g. 6 = 1+2+3).
- Real numbers = rationals + irrationals; they cover the entire number line.
Types of Numbers โ Flashcards (SSC CGL)
Cover the answer, recall, then check. 7 cards on the must-know Types of Numbers facts for SSC CGL.
Q1. What is the only even prime number?
A1. 2.
Q2. Is 1 a prime number?
A2. No โ 1 is neither prime nor composite.
Q3. Define a rational number.
A3. A number expressible as p/q where p and q are integers and q โ 0.
Q4. Give an example of an irrational number.
A4. โ2 (or ฯ) โ it cannot be written as a ratio of two integers.
Q5. What are co-prime numbers?
A5. Two numbers whose HCF is 1 (e.g. 8 and 9).
Q6. How many factors does a prime number have?
A6. Exactly two โ 1 and the number itself.
Q7. Why is 6 called a perfect number?
A7. Because the sum of its proper divisors (1+2+3) equals 6.
Types of Numbers โ Formula Sheet
Key formulas
- Classification: natural โ whole โ integer โ rational โ real; irrationals are non-terminating non-repeating.
- Prime: exactly two factors; composite: more than two; 1 is neither.
- Co-prime: HCF = 1. Perfect number = sum of proper divisors (e.g. 6, 28).
- Sum of first n naturals n(n+1)/2; first n odd numbers = nยฒ; first n even = n(n+1).
- Rational number lies between any two reals; between a and b: (a+b)/2.
- A number is rational iff its decimal terminates or repeats.
- โ- Prime has exactly 2 factors; 1 is neither prime nor composite.
- โ- Co-prime โ HCF = 1.
- โ- Sum of first n odd numbers = nยฒ.
- โ- Rational โ terminating or repeating decimal.
Usage: identify the number type first โ it fixes which divisibility/sum rules apply.