Sieve of Eratosthenes Quiz: Explore How Prime Numbers Are Found

The prime numbers in the range 1 to 10 are 2, 3, 5, and 7. That gives exactly 4 primes. The number 1 is excluded because it has only one factor. The numbers 4, 6, 8, 9, and 10 are all composite: 4 equals 2 multiplied by 2, 6 equals 2 multiplied by 3, 8 equals 2 multiplied by 4, 9 equals 3 multiplied by 3, and 10 equals 2 multiplied by 5.

The number 25 equals 5 multiplied by 5, giving it three factors: 1, 5, and 25. Because it has more than two factors it is composite and is crossed out when the sieve eliminates multiples of 5. The numbers 23, 31, and 37 are all prime because none of them can be divided evenly by 2, 3, or 5, and each has exactly two factors.

After the first step of the sieve, every even number greater than 2 has been crossed out. What remains is 2 and all the odd numbers in the range. However, not all odd numbers are prime. For example, 9 equals 3 multiplied by 3 and 15 equals 3 multiplied by 5, both odd but composite. Odd numbers that are multiples of 3, 5, or 7 will be eliminated in later steps.

The answer is True. Two consecutive integers always include one even and one odd number. Since the only even prime is 2, the only way two consecutive integers can both be prime is if one of them is 2. The pair 2 and 3 satisfies this because 2 is even and prime and 3 is odd and prime. No other consecutive pair can both be prime because the even one would be divisible by 2.

Only the number 25 is newly crossed out during the step for 5. The number 10 equals 2 multiplied by 5 and was already eliminated as a multiple of 2. The number 20 equals 4 multiplied by 5 and was also already eliminated as a multiple of 2. The number 30 equals 2 multiplied by 15 and was already eliminated as a multiple of 2. The number 25 equals 5 multiplied by 5 and has no earlier prime factor, making it the only new elimination in this step.

The number 1 is neither prime nor composite. A prime number must have exactly two distinct factors: 1 and itself. The number 1 has only one factor, which is 1 alone. This disqualifies it from being prime. It is also not composite because composite numbers must have more than two factors. The sieve excludes 1 entirely before the process begins.

The prime numbers in the range 11 to 20 are 11, 13, 17, and 19. That gives exactly 4 primes. The numbers 12, 14, 15, 16, 18, and 20 are all composite: 12 equals 4 multiplied by 3, 14 equals 2 multiplied by 7, 15 equals 3 multiplied by 5, 16 equals 2 multiplied by 8, 18 equals 2 multiplied by 9, and 20 equals 4 multiplied by 5.

The answer is True. Every prime number greater than 2 is odd, because any even number greater than 2 is divisible by 2 and therefore has more than two factors, making it composite. The number 2 is the single exception: it is even and also prime because its only factors are 1 and 2. This makes 2 the only even prime and therefore the only prime that is not odd.

The square root of 37 is approximately 6.08, so only the primes up to 6 need to be checked: 2, 3, and 5. The number 37 is odd, not divisible by 3, and does not end in 0 or 5. Because it passes all three checks it is confirmed prime. Any number left uncrossed by the sieve is prime for exactly this reason — all relevant composite factors have already been eliminated.

The numbers 11 and 13 are both prime because neither is divisible by 2, 3, or 5. The numbers 17 and 19 are both prime for the same reason. These are examples of twin primes, pairs that differ by exactly 2. The number 21 equals 3 multiplied by 7 and is composite, so option C does not qualify. The number 27 equals 3 multiplied by 9 and is also composite, eliminating option D.

The range 1 to 20 contains 8 primes: 2, 3, 5, 7, 11, 13, 17, and 19. The number 1 is neither prime nor composite. That leaves 20 minus 8 minus 1 equals 11 composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, and 20. Composites are numbers greater than 1 that have more than two factors and are crossed out during the sieve.

The answer is True. The square root of 53 is approximately 7.28. This means any composite factor of 53 must be 7 or smaller. The primes up to 7 are 2, 3, 5, and 7. Since 53 is odd, not divisible by 3, does not end in 0 or 5, and 53 divided by 7 gives approximately 7.57 with no whole number result, it passes all checks. No prime larger than 7 needs to be tested, confirming 53 is prime.

The number 53 is not divisible by 2, 3, 5, or 7, and since the square root of 53 is less than 8, no further checking is needed. It is therefore prime. The number 51 equals 3 multiplied by 17, 55 equals 5 multiplied by 11, and 57 equals 3 multiplied by 19, so all three are composite and would be crossed out by the sieve.

The prime numbers in the range 21 to 30 are 23 and 29. That gives exactly 2 primes. The number 21 equals 3 multiplied by 7, 22 is even, 24 is even, 25 equals 5 multiplied by 5, 26 is even, 27 equals 3 multiplied by 9, 28 is even, and 30 is even, so all are composite. This range has fewer primes than 1 to 10 or 11 to 20, showing how prime frequency decreases in higher ranges.

The number 49 equals 7 multiplied by 7, giving it three distinct factors: 1, 7, and 49. Because it has more than two factors it is composite. The sieve crosses it out during the step for prime 7. The fact that 49 is odd and does not end in 0 or 5 only rules out divisibility by 2 and 5. Without checking divisibility by 7, a number cannot be confirmed prime based on those two checks alone.

The answer is True. In the range 1 to 50, there are 15 prime numbers and 34 composite numbers, with 1 being neither prime nor composite. Composites significantly outnumber primes. This pattern becomes more pronounced as the range increases because larger numbers have more small prime divisors available to eliminate them, meaning a greater share of numbers in any range will be composite.

A prime number must have exactly two factors: 1 and itself. Any even number greater than 2 is divisible by 2, which means it has at least three factors: 1, 2, and itself. This immediately disqualifies it from being prime. Since every even number greater than 2 fails this check, the only even number that can be prime is 2 itself.

The primes from 51 to 70 are 53, 59, 61, and 67. The number 57 equals 3 multiplied by 19 and is composite, eliminating option A. The number 63 equals 9 multiplied by 7 and is composite, eliminating option C. The number 55 equals 5 multiplied by 11 and is composite, eliminating option D. Each number in option B is verified prime by checking divisibility up to its square root.

The answer is True. A prime number must have exactly two factors: 1 and itself. Any number with exactly three factors has one additional divisor beyond 1 and itself, which means it is composite. Numbers with exactly three factors are perfect squares of prime numbers, such as 4 which has factors 1, 2, and 4, or 9 which has factors 1, 3, and 9. None of these qualify as prime.

There are 15 primes in 1 to 50 and 25 primes in 1 to 100. The count increases from 15 to 25, confirming more primes exist in the larger range. However the count does not double: doubling 15 would give 30, but only 25 primes exist up to 100. This is because primes become less frequent as numbers increase, so the additional range of 51 to 100 contributes only 10 new primes rather than 15.