Sieve of Eratosthenes Find All Primes in DSA C# - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main idea behind the Sieve of Eratosthenes?

It is a method to find all prime numbers up to a certain limit by repeatedly marking the multiples of each prime number starting from 2.

intermediate

Why do we start marking multiples from the square of the current prime in the Sieve of Eratosthenes?

Because all smaller multiples of the prime have already been marked by smaller primes, so starting from the square avoids redundant work.

beginner

In the Sieve of Eratosthenes, what does a 'true' value in the boolean array represent?

It means the number at that index is currently considered prime (not marked as a multiple of any smaller prime).

intermediate

What is the time complexity of the Sieve of Eratosthenes algorithm?

The time complexity is approximately O(n log log n), which is efficient for finding primes up to large numbers.

beginner

How does the Sieve of Eratosthenes help in real-life scenarios?

It helps quickly find prime numbers which are important in cryptography, hashing, and number theory applications.

What is the first prime number used in the Sieve of Eratosthenes?

A 1

B 0

C 2

D 3

The sieve starts with 2, the smallest prime number.

When marking multiples of a prime p, from which number do we start marking?

A p * p

B 2 * p

C p + 1

D 1

We start marking from p squared to avoid marking multiples already handled by smaller primes.

What data structure is commonly used to keep track of prime candidates in the sieve?

A Boolean array

B Queue

C Stack

D Linked list

A boolean array marks whether each number is prime (true) or not (false).

Which of these numbers will be marked as not prime when processing prime 3?

A 6

B 9

C 12

D All of the above

All multiples of 3 (6, 9, 12, etc.) are marked as not prime.

What is the stopping condition for the outer loop in the sieve?

A When current prime > n

B When current prime > sqrt(n)

C When current prime > n/2

D When current prime > 1

We only need to check primes up to the square root of n.

Explain how the Sieve of Eratosthenes finds all prime numbers up to a given number n.

Think about crossing out multiples step by step.

Describe why the Sieve of Eratosthenes is more efficient than checking each number individually for primality.

Focus on how it reduces repeated checks.

Practice

1. What is the main purpose of the Sieve of Eratosthenes algorithm?

easy

A. To sort an array of numbers

B. To find the greatest common divisor of two numbers

C. To find all prime numbers up to a given limit

D. To reverse a string

Solution

1. Step 1: Understand the algorithm's goal

   The Sieve of Eratosthenes is designed to identify prime numbers efficiently.

2. Step 2: Recognize the output

   It marks multiples of primes as non-prime to find all primes up to a limit.

3. Final Answer:

   To find all prime numbers up to a given limit -> Option C

4. Quick Check:

   Sieve of Eratosthenes = Find primes [OK]

Hint: Remember: Sieve filters out non-primes to list primes [OK]

Common Mistakes:

• Confusing it with sorting algorithms
• Thinking it finds factors instead of primes
• Assuming it works on strings

2. Which of the following C# code snippets correctly initializes a boolean array for the sieve to find primes up to n?

easy

A. bool[] isPrime = new bool[n + 1]; for(int i = 0; i <= n; i++) isPrime[i] = false;

B. bool[] isPrime = new bool[n + 1]; for(int i = 2; i <= n; i++) isPrime[i] = true;

C. bool[] isPrime = new bool[n]; for(int i = 0; i < n; i++) isPrime[i] = false;

D. bool[] isPrime = new bool[n]; for(int i = 2; i < n; i++) isPrime[i] = true;

Solution

1. Step 1: Check array size and initialization

   The array must be of size n+1 to include index n, and primes start from 2.

2. Step 2: Verify initialization values

   All entries from 2 to n should be set to true initially to mark potential primes.

3. Final Answer:

   bool[] isPrime = new bool[n + 1]; for(int i = 2; i <= n; i++) isPrime[i] = true; -> Option B

4. Quick Check:

   Array size n+1 and true from 2 to n [OK]

Hint: Array size n+1 and true from 2 to n for primes [OK]

Common Mistakes:

• Using array size n instead of n+1
• Initializing all values to false
• Starting initialization from 0 or 1

3. What will be the output list of primes when running the Sieve of Eratosthenes for n = 10?

medium

A. [2, 3, 5, 7, 9]

B. [1, 2, 3, 5, 7]

C. [2, 3, 4, 5, 7]

D. [2, 3, 5, 7]

Solution

1. Step 1: Identify primes up to 10

   Primes are numbers greater than 1 with no divisors other than 1 and themselves: 2, 3, 5, 7.

2. Step 2: Exclude non-primes

   Numbers 1, 4, 6, 8, 9, 10 are not prime.

3. Final Answer:

   [2, 3, 5, 7] -> Option D

4. Quick Check:

   Primes up to 10 = [2, 3, 5, 7] [OK]

Hint: Remember primes > 1 and exclude multiples [OK]

Common Mistakes:

• Including 1 as prime
• Including even numbers > 2
• Including non-prime odd numbers like 9

4. Consider this code snippet for the sieve:

for (int i = 2; i * i <= n; i++) {
  if (isPrime[i]) {
    for (int j = i * 2; j <= n; j += i) {
      isPrime[j] = false;
    }
  }
}

Which of the following improves the efficiency of this code?

medium

A. The inner loop should start from i * i, not i * 2

B. The outer loop condition should be i <= n, not i * i <= n

C. The inner loop increments should be j += 1, not j += i

D. The isPrime array should be initialized with false, not true

Solution

1. Step 1: Understand sieve optimization

   Marking multiples starts from i squared because smaller multiples were marked by smaller primes.

2. Step 2: Identify suboptimal start of inner loop

   Starting from i*2 repeats marking multiples already handled; correct start is i*i.

3. Final Answer:

   The inner loop should start from i * i, not i * 2 -> Option A

4. Quick Check:

   Start inner loop at i*i for efficiency [OK]

Hint: Start marking multiples from i squared, not double i [OK]

Common Mistakes:

• Starting inner loop at i*2 causing redundant work
• Changing outer loop condition incorrectly
• Incorrect increment steps in inner loop

5. You want to find all prime numbers up to n = 30 using the Sieve of Eratosthenes. After running the sieve, which of the following lists correctly shows the primes and their count?

hard

A. Primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29], Count: 10

B. Primes: [2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29], Count: 11

C. Primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29], Count: 11

D. Primes: [1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29], Count: 11

Solution

1. Step 1: List primes up to 30

   Primes are numbers >1 with no divisors except 1 and themselves: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

2. Step 2: Count and verify correctness

   Count is 10; numbers like 9, 25, and 1 are not prime and should be excluded.

3. Final Answer:

   Primes: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29], Count: 10 -> Option A

4. Quick Check:

   Correct primes and count up to 30 [OK]

Hint: Exclude 1 and non-primes like 9, 25; count primes carefully [OK]

Common Mistakes:

• Including 1 as prime
• Including composite numbers like 9 or 25
• Miscounting the total primes