Sieve Comparison Quiz: Test Your Knowledge on Different Sieve Ranges

The Sieve of Eratosthenes is a systematic method for identifying all prime numbers up to a given limit. Starting at 2, the method circles each prime and crosses out all of its multiples. Any number that is never crossed out by the end of the process has no factors other than 1 and itself, confirming that it is prime.

The answer is False. A prime number must have exactly two distinct factors: 1 and itself. The number 1 has only one factor, which is 1 alone. Since it does not satisfy the two-factor requirement, 1 is not considered prime. The Sieve of Eratosthenes excludes 1 from the process entirely, and the first number circled is always 2.

Every even number greater than 2 is divisible by 2, which means it has at least three factors and is therefore composite. The number 2 itself has only two factors, 1 and 2, which qualifies it as prime. No other even number can meet this standard. Being the only even prime makes 2 unique among all prime numbers.

The multiples of 2 that are greater than 2 and within the range 1 to 30 are 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, and 30. Counting them gives 14 numbers. The number 2 itself is circled as prime and is not crossed out. Every other even number in the range is eliminated in this first step of the sieve.

The answer is True. After crossing out all multiples of 2, such as 4, 6, 8, and 10, the next number that has not been crossed out is 3. Because no earlier step has eliminated it, 3 has no known prime factors smaller than itself. It is therefore prime and gets circled. The process then continues by crossing out all multiples of 3 greater than 3.

The number 9 equals 3 multiplied by 3, which gives it three distinct factors: 1, 3, and 9. Because it has more than two factors, it is composite. When the sieve reaches 3 and begins crossing out its multiples, 9 is one of the first new numbers eliminated. Being odd does not protect a number from being crossed out if it is a multiple of another prime.

The primes in the range 1 to 20 include 2, 3, 5, 7, 11, 13, 17, and 19. The number 9 is not prime because it equals 3 multiplied by 3 and is crossed out when multiples of 3 are eliminated. The numbers 7, 13, and 19 are each divisible only by 1 and themselves, so they remain uncrossed throughout the entire sieve process and are confirmed as prime.

Any composite number has at least one factor less than or equal to its square root. Since the square root of 100 is exactly 10, any composite number up to 100 must have a prime factor of 10 or less. The primes up to 10 are 2, 3, 5, and 7. Once multiples of all four are crossed out, every remaining uncrossed number is confirmed prime without any further checking needed.

The answer is True. When the sieve reaches 7, all multiples of 7 smaller than 49 have already been crossed out in earlier steps. For example, 14 was removed as a multiple of 2, 21 as a multiple of 3, and 35 as a multiple of 5. The first multiple of 7 not yet eliminated within 1 to 50 is 7 multiplied by 7, which equals 49.

The prime numbers between 31 and 50 are 31, 37, 41, 43, and 47. The number 33 equals 3 multiplied by 11, 39 equals 3 multiplied by 13, and 45 equals 5 multiplied by 9, so all three are composite and crossed out. Every number in the correct list has only two factors, 1 and itself, and remains uncrossed throughout the entire sieve process.

The prime numbers from 1 to 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47. Counting all of them gives exactly 15 primes. These are all the numbers in that range left uncrossed after eliminating multiples of 2, 3, 5, and 7. Any other number in the range has at least one factor from that group and is therefore composite.

In the range 1 to 30, there are 10 primes out of 30 numbers. In the range 31 to 50, there are only 5 primes out of 20 numbers. As numbers grow larger, more small primes such as 2, 3, 5, and 7 are available to eliminate candidates as multiples. This means a greater proportion of larger numbers get crossed out, leaving fewer primes behind.

The answer is True. Working backwards from 50, the number 49 equals 7 multiplied by 7 and is composite. The number 48 is even and therefore composite. The number 47 is not divisible by 2, 3, 5, or 7, and since those are all the primes needed to check up to its square root, no further checking is required. This confirms that 47 is prime and is the largest prime below 50.

The answer is False. The range 1 to 30 contains 10 prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. The range 31 to 50 contains only 5 primes: 31, 37, 41, 43, and 47. Primes become less frequent as numbers increase because larger numbers have more small prime divisors available to eliminate them as composites during the sieve.

The student checked only divisibility by 2 and 5. To confirm a number is prime, all prime factors up to its square root must be tested. The square root of 51 is approximately 7.1, so 2, 3, 5, and 7 must each be checked. Dividing 51 by 3 gives exactly 17 with no remainder, which means 51 is composite with three factors: 1, 3, and 17.

The sieve starts at 2 and works upward, making option A correct. Any uncrossed number has no prime factors below it, confirming it is prime, making option B correct. Option C is incorrect because the sieve moves from smallest to largest and eliminates entire groups of multiples at once rather than checking numbers individually. Option D is correct because no prime larger than 7 is needed when the upper limit is 50.

The complete and correct list of primes from 1 to 30 is 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29, giving exactly 10 primes. Option A incorrectly includes 9, which equals 3 multiplied by 3. Option B incorrectly includes 1, which is not prime by definition. Option C incorrectly includes 27, which equals 3 multiplied by 9. Only option D contains all 10 primes with no errors.

The answer is True. The sieve works by taking each newly circled prime and crossing out all of its multiples that are greater than itself. A number is crossed out because it shares a factor with a smaller prime, which makes it composite. Numbers that do not share a factor with any previously identified prime remain uncrossed and are eventually circled as prime themselves.

Completing the sieve for the range 1 to 30 identifies 10 prime numbers. Completing the sieve for the range 1 to 50 identifies 15 prime numbers. Subtracting 10 from 15 gives a difference of 5. The five additional primes found in the extended range are 31, 37, 41, 43, and 47, each of which falls between 31 and 50 and remains uncrossed after all multiples are eliminated.

If a number is composite, it can be written as a product of two factors. At least one of those factors must be less than or equal to the square root of the number, because if both factors were larger, their product would exceed the number itself. This means every composite in the range is already crossed out before the sieve reaches a prime larger than the square root of the upper limit.