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Prime Number Series

Introduction

Prime Number Series use prime numbers (numbers greater than 1 with no positive divisors other than 1 and themselves). These series appear often in reasoning tests to check number-sense and quick divisibility checks.

Pattern: Prime Number Series

Pattern

The key idea: Each term is a prime number - the next term is the next prime in ascending order.

Definition: A prime is an integer > 1 that has no positive integer divisors other than 1 and itself.

Common Methods to Identify Next Prime:
• Check successive integers after the last term and test for divisibility by primes ≤ √n (trial division).
• Use elimination: skip even numbers (except 2) and numbers ending with 5 (except 5).
• For small series, memorise the first few primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

Notes & Patterns:
• 2 is the only even prime; all other primes are odd.
• Gaps between consecutive primes vary (prime gaps).
• Special subpatterns: twin primes (pairs like 11 & 13), prime squares rarely used as prime terms.

Step-by-Step Example

Question

Find the next term in the series: 2, 3, 5, 7, 11, 13, ?

Solution

  1. Step 1: Observe the given terms

    The terms listed are consecutive prime numbers: 2, 3, 5, 7, 11, 13.

  2. Step 2: Check the next integer after 13

    Test 14 (even → not prime), 15 (divisible by 3 → not prime), 16 (even → not prime), 17 (trial division: not divisible by 2,3,5).

  3. Step 3: Confirm primality of 17

    √17 ≈ 4.12 - test divisibility by primes ≤ 4 (2 and 3). 17 is not divisible by 2 or 3, so 17 is prime.

  4. Final Answer:

    17

  5. Quick Check:

    The next prime after 13 is 17 (14-16 are composite) ✅

Quick Variations

1. Prime pairs (twin primes): e.g., (11, 13), (17, 19).

2. Prime gaps: series may skip primes to form a pattern (e.g., every second prime).

3. Mixed patterns: primes combined with arithmetic shifts (e.g., prime + 2, prime × 2 patterns).

Trick to Always Use

  • Step 1 → Memorise small primes up to at least 50 for quick lookup.
  • Step 2 → For a candidate number n, test divisibility only by primes ≤ √n (2,3,5,7,11...).

Summary

Summary

  • Recognise prime series by checking each term has no divisors except 1 and itself.
  • Skip checking even numbers (except 2) and multiples of 5 (except 5) to narrow candidates.
  • Use trial division up to √n to confirm primality for small n quickly.
  • Memorise the first 15-20 primes for speed in tests.

Example to remember:
Series: 11, 13, 17, 19 → next = 23 (primes in ascending order)

Practice

(1/5)
1. Find the next term in the series: 2, 3, 5, 7, 11, ?
easy
A. 12
B. 13
C. 14
D. 17

Solution

  1. Step 1: Identify the pattern

    The terms are consecutive prime numbers: 2, 3, 5, 7, 11.

  2. Step 2: Find the next prime

    The next prime after 11 is 13.

  3. Final Answer:

    13 → Option B

  4. Quick Check:

    Prime sequence continues correctly.
Hint: Memorize first 15 primes for quick spotting.
Common Mistakes: Choosing composite numbers like 12 or 14.
2. Find the missing term: 5, 11, 17, 23, ?
easy
A. 27
B. 29
C. 31
D. 33

Solution

  1. Step 1: Observe pattern

    This is a prime series with constant difference +6.

  2. Step 2: Compute next term

    23 + 6 = 29.

  3. Step 3: Verify primality

    29 is prime.

  4. Final Answer:

    29 → Option B

  5. Quick Check:

    All terms maintain +6 increment and stay prime.
Hint: Prime + even gap often remains prime.
Common Mistakes: Choosing 31 without confirming gap.
3. Identify the next prime: 11, 13, 17, 19, ?
easy
A. 21
B. 23
C. 25
D. 27

Solution

  1. Step 1: Recognize the pattern

    The series lists consecutive primes.

  2. Step 2: Check primes after 19

    20 ❌ (even), 21 ❌ (3×7), 22 ❌ (even), 23 ✔ (prime).

  3. Final Answer:

    23 → Option B

  4. Quick Check:

    23 is next prime after 19.
Hint: Skip even numbers & multiples of 3 when checking primality.
Common Mistakes: Selecting 21 or 25 assuming +2 rule.
4. Find the next number: 13, 17, 19, 23, 29, ?
medium
A. 31
B. 33
C. 35
D. 37

Solution

  1. Step 1: Read the pattern

    The numbers are consecutive primes.

  2. Step 2: Next prime after 29

    30 ❌, 31 ✔.

  3. Final Answer:

    31 → Option A

  4. Quick Check:

    Prime list confirms 31 is next.
Hint: Memorize primes until 50 for quick solving.
Common Mistakes: Choosing 37 instead of the immediate next prime.
5. Find the next prime term: 17, 29, 41, 53, ?
medium
A. 59
B. 61
C. 67
D. 71

Solution

  1. Step 1: Identify the rule

    The series increases by a constant arithmetic gap of +12: 17 → 29 (+12), 29 → 41 (+12), 41 → 53 (+12).

  2. Step 2: Apply the +12 rule

    Next value = 53 + 12 = 65.

  3. Step 3: Check primality

    65 is composite (65 = 5 × 13), so it cannot be used in a prime series.

  4. Step 4: Apply the fallback rule

    When the +12 result is composite, select the next immediate prime after it. The next prime after 65 is 67.

  5. Final Answer:

    67 → Option C

  6. Quick Check:

    Pattern: +12 each step → adjust to next prime if composite → 65 (composite) → 67 (prime) ✅
Hint: If a fixed gap leads to a composite number, move to the next immediate prime to preserve the prime-series structure.
Common Mistakes: Stopping at 65 without checking primality or switching to consecutive-prime logic incorrectly.

Mock Test

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