Prime and Composite Numbers
Pointwise AISSEE Class 6 Mathematics guide for Prime and Composite Numbers (Topic 19). Master prime number definitions, twin primes, co-primes, and divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10, 11.
Overview & Fundamental Concept
Pointwise AISSEE Class 6 Mathematics guide for Prime and Composite Numbers (Topic 19). Master prime number definitions, twin primes, co-primes, and divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10, 11.
Official Syllabus Coverage Breakdown
| Syllabus Item | Prescribed Scope | Exam Relevance & Pedagogical Objective |
|---|---|---|
| Item 19 | Prime numbers, composite numbers, tests of divisibility (by 2, 3, 4, 5, 6, 8, 9, 10, 11), twin primes, co-primes. | Master prime/composite taxonomy, twin prime pairs, co-prime definitions, and rapid divisibility test algorithms (2 to 11) for AISSEE Class VI. |
1. Prime vs. Composite Taxonomy & Fundamental Axioms
- Prime Numbers: A natural number strictly greater than 1 that has exactly two distinct factors: $1$ and the number itself. Examples: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, \dots$
- Composite Numbers: A natural number having more than two factors. Examples: $4, 6, 8, 9, 10, 12, 14, 15, 16, \dots$
- The Special Status of 1: The number 1 is NEITHER prime NOR composite (because it has only one single factor: 1).
- The Even Prime Axiom: The number 2 is the smallest prime number and the ONLY even prime number. Every other prime number is odd.
- Smallest Composite Numbers: The smallest composite number is 4. The smallest odd composite number is 9.
- Prime Count Milestone: There are exactly 25 prime numbers between 1 and 100.
| Category | Definition | Smallest Value | Examples |
|---|---|---|---|
| Prime Number | Exactly 2 distinct factors (1 and self) | 2 (only even prime) | 2, 3, 5, 7, 11, 13, 17, 19 |
| Composite Number | More than 2 factors | 4 (even), 9 (odd) | 4, 6, 8, 9, 10, 12, 14 |
| Neither Prime nor Composite | Has only 1 factor | 1 | 1 |
| Twin Primes | Two primes differing by exactly 2 | (3, 5) | (3, 5), (5, 7), (11, 13), (17, 19) |
| Co-Prime Numbers | Two numbers having HCF = 1 | (1, 2) | (8, 15), (14, 15) |
Cadet Mnemonic • Prime Reality Anchor
Never forget the core properties:
- 1 is Neither (Lonely 1)
- 2 is the ONLY Even Prime (King 2)
- 4 is the Smallest Composite
2. Twin Primes & Co-Prime Numbers
- Twin Primes: Pairs of prime numbers whose difference is exactly 2: $p_2 - p_1 = 2$.
- All 8 Pairs of Twin Primes Below 100:
$$(3, 5), \; (5, 7), \; (11, 13), \; (17, 19), \; (29, 31), \; (41, 43), \; (59, 61), \; (71, 73)$$ - Co-Prime (Relatively Prime) Numbers: Two numbers are co-prime if their Highest Common Factor is 1 ($\text{HCF} = 1$).
- Key Co-Prime Properties:
• Two numbers do NOT need to be prime individually to be co-prime (e.g., $8$ and $15$ are both composite, but $\text{HCF}(8, 15) = 1$, so they are co-prime).
• Any two consecutive natural numbers ($n, n+1$) are ALWAYS co-prime.
• Any two distinct prime numbers are ALWAYS co-prime.
AISSEE Trap Alert • Twin Primes vs Co-Primes Confusion
Twin primes MUST both be prime numbers (e.g. 5 and 7). Co-primes do NOT have to be prime numbers (e.g. 8 and 9 are co-prime, but neither is a prime number!).
3. Comprehensive Tests of Divisibility (2 to 11)
- Divisibility by 2: Units digit is even ($0, 2, 4, 6, 8$).
- Divisibility by 3: Sum of all digits is divisible by 3.
- Divisibility by 4: Number formed by the last two digits is divisible by 4 (or ends in 00).
- Divisibility by 5: Units digit is 0 or 5.
- Divisibility by 6: Number is divisible by BOTH 2 and 3.
- Divisibility by 8: Number formed by the last three digits is divisible by 8 (or ends in 000).
- Divisibility by 9: Sum of all digits is divisible by 9.
- Divisibility by 10: Units digit is strictly 0.
- Divisibility by 11: The difference between the sum of digits at odd places (from right) and the sum of digits at even places (from right) is either 0 or a multiple of 11.
| Divisor | Divisibility Test Rule | Verification Example |
|---|---|---|
| 2 | Last digit is 0, 2, 4, 6, 8 | 4,786 (ends in 6 → Divisible) |
| 3 | Sum of digits is multiple of 3 | 5,472 ($5+4+7+2 = 18$ → Divisible) |
| 4 | Last 2 digits divisible by 4 | 3,528 (28 ÷ 4 = 7 → Divisible) |
| 6 | Divisible by both 2 and 3 | 738 (Even & $7+3+8=18$ → Divisible) |
| 8 | Last 3 digits divisible by 8 | 5,128 (128 ÷ 8 = 16 → Divisible) |
| 9 | Sum of digits is multiple of 9 | 8,361 ($8+3+6+1 = 18$ → Divisible) |
| 11 | |Sum(Odds) − Sum(Evens)| = 0 or 11k | 9,0728: $(8+7+9) - (2+0) = 24-2 = 22$ (Divisible) |
Exam Trick • Divisibility by 11 Alternating Sum
For 1,331: Odd places = 1 + 3 = 4. Even places = 3 + 1 = 4. Difference = 4 − 4 = 0. Divisible by 11!
Visual Concept Blueprint • Vector Architecture
Speed Shortcuts & Mental Calculation Strategies
Master these high-yield time-saving heuristics curated by top AISSEE scorers to eliminate calculation bottlenecks and answer questions in under 45 seconds.
Shortcut Technique 1: Prime Identification Test (Up to √N)
To test if $N$ is prime: Check divisibility by prime numbers up to $\sqrt{N}$. For 97: $\sqrt{97} < 10$. Test primes 2, 3, 5, 7. Since none divides 97, 97 is prime!
Shortcut Technique 2: Divisibility by Composite Numbers
To check divisibility by composite $k = a \times b$ where $a$ and $b$ are co-prime: Test divisibility by $a$ and $b$ separately (e.g., for 12, test 3 and 4; for 15, test 3 and 5).
Shortcut Technique 3: 25 Primes Under 100 Anchor
Memorize prime counts in tens: 1-10: 4 (2,3,5,7); 11-20: 4 (11,13,17,19); 21-30: 2 (23,29); 31-40: 2 (31,37); 41-50: 3 (41,43,47). Total up to 50 = 15 primes.
Interactive 3D Revision Flashcards Deck
Click any card to flip and reveal the key exam rule or topper secret.
REVISION FLASHCARD
What is the core rule & definition for Prime and Composite Numbers?
CADET MEMORY KEY
Divisibility by 3 & 9: \sum \text{digits} = 3k \text{ or } 9k
REVISION FLASHCARD
Cadet Shortcut: How to solve Prime and Composite Numbers in under 30 seconds?
CADET MEMORY KEY
Fast MCQ Elimination: Check the last digit (unit digit) and estimate the order of magnitude to eliminate 2 distractors in under 10 seconds.
REVISION FLASHCARD
Exam Trap Alert: What pitfall do most candidates fall into?
CADET MEMORY KEY
Unique Even Prime: 2 is the only even prime number; all other even numbers are composite.
REVISION FLASHCARD
Rapid Memory Anchor for AISSEE Hall Recall:
CADET MEMORY KEY
Special Status of 1: 1 is NEITHER prime NOR composite because it has only 1 factor.
REVISION FLASHCARD
Is 91 a prime number?
CADET MEMORY KEY
No! 91 is a composite number because 91 = 7 × 13 (it has factors 1, 7, 13, 91).
Common Traps & Examination Pitfalls
Avoid the Speed-Guess Trap: Under AISSEE pressure, candidates frequently misread units (e.g., mixing metres and centimetres, or per hour and per second) and confuse complementary ($90^\circ$) with supplementary ($180^\circ$) angles. Read every stem carefully and check units before selecting your OMR option.
Key Identities & Formulas
Divisibility by 11 Rule
$$|\sum d_{\text{odd}} - \sum d_{\text{even}}| = 11k$$
Where k is 0, 1, 2, ...
Twin Prime Gap
$$p_2 - p_1 = 2$$
Constant gap of 2 between twin prime pairs.
Prime Count up to 100
$$\pi(100) = 25$$
Total 25 prime numbers below 100.
Step-by-Step Solved Examination Models
Solved Example 1
Model Problem
Which of the following is the only EVEN prime number?
Detailed Solution:
2 is the only even prime number. Every other even number is divisible by 2 and therefore composite.
Solved Example 2
Model Problem
How many prime numbers are there between 1 and 50?
Detailed Solution:
The 15 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
Solved Example 3
Model Problem
What least non-zero digit should replace * in the number 5*342 so that the resulting number is completely divisible by 9?
Detailed Solution:
1. A number is divisible by 9 if the sum of all its digits is a multiple of 9.
2. Sum of known digits $= 5 + * + 3 + 4 + 2 = 14 + *$.
3. The next multiple of 9 greater than 14 is 18.
4. $14 + * = 18 \implies * = 18 - 14 = 4$.
5. Replacing * with 4 gives 54,342, whose digit sum is $5+4+3+4+2 = 18$ (divisible by 9). Correct Option is (A).
2. Sum of known digits $= 5 + * + 3 + 4 + 2 = 14 + *$.
3. The next multiple of 9 greater than 14 is 18.
4. $14 + * = 18 \implies * = 18 - 14 = 4$.
5. Replacing * with 4 gives 54,342, whose digit sum is $5+4+3+4+2 = 18$ (divisible by 9). Correct Option is (A).
AISSEE High-Yield Practice Drills (10 Questions)
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Cadet Streak 🔥
Question 1 of 10
Exam Standard
What is the smallest composite number?
Correct Answer: Option (A)
1 is neither prime nor composite. 2 and 3 are prime. 4 is the smallest composite number (factors: 1, 2, 4).
Question 2 of 10
Exam Standard
Which number is NEITHER prime NOR composite?
Correct Answer: Option (A)
1 is unique: it has only one positive divisor (itself), so it is neither prime nor composite.
Question 3 of 10
Exam Standard
Which pair of numbers represents 'Twin Primes'?
Correct Answer: Option (A)
Twin primes are pairs of prime numbers that differ by exactly 2. $13 - 11 = 2$.
Question 4 of 10
Exam Standard
What is the prime factorisation of 72?
Correct Answer: Option (A)
$72 = 8 \times 9 = 2^3 \times 3^2$.
Question 5 of 10
Exam Standard
What is the smallest odd prime number?
Correct Answer: Option (A)
2 is the smallest prime (but even). 3 is the smallest odd prime.
Question 6 of 10
Exam Standard
Two numbers whose Highest Common Factor (HCF) is 1 are called:
Correct Answer: Option (A)
Two numbers having no common positive divisor other than 1 are co-prime (e.g. 8 and 15).
Question 7 of 10
Exam Standard
Which of the following numbers is a prime number?
Correct Answer: Option (A)
$91 = 7 \times 13$, $87 = 3 \times 29$, $93 = 3 \times 31$. Only 97 has no factors other than 1 and 97 (largest 2-digit prime).
Question 8 of 10
Exam Standard
How many prime numbers are there in total between 1 and 100?
Correct Answer: Option (A)
There are exactly 25 prime numbers between 1 and 100.
Question 9 of 10
Exam Standard
The sum of two consecutive prime numbers is 30. What are the numbers?
Correct Answer: Option (A)
13 and 17 are consecutive prime numbers and $13 + 17 = 30$.
Question 10 of 10
Exam Standard
What is the smallest 3-digit prime number?
Correct Answer: Option (A)
100 is composite ($10 \times 10$). 101 cannot be divided by 2, 3, 5, or 7, making it the smallest 3-digit prime.
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Last-Minute Cadet Revision Matrix
Rapid recall checkpoints to review 24 hours before stepping into the examination hall.
| Concept Checkpoint | High-Yield Exam Rule & Memory Anchor |
|---|---|
| Number 1 | 1 is NEITHER prime nor composite. |
| Number 2 | 2 is the smallest prime number and the ONLY even prime. |
| Twin Primes | Twin primes are prime pairs with a difference of 2 (e.g. 11 and 13). |
| Divisibility by 9 | Sum of digits must be a multiple of 9. |
Frequently Asked Questions
How many questions from Prime and Composite Numbers appear in the AISSEE Class VI exam?
Typically, 2 to 3 questions directly test Prime and Composite Numbers in the 50-question Mathematics paper, contributing 6 to 9 marks towards the 150-mark section total.
Is there any negative marking for incorrect answers in AISSEE Class VI?
No. The National Testing Agency (NTA) does not apply negative marking in the AISSEE Class VI entrance examination. Candidates should attempt every single question.
What is the best way to revise Prime and Composite Numbers before the exam?
Review the high-yield formulas, practice the 10 interactive drill MCQs on this page, and memorize the revision checkpoints matrix.