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In the Sieve of Eratosthenes for numbers less than 100, explain why, after crossing out all the multiples of 2, 3, 5, and 7, the remaining numbers are prime.



Choose the correct answer below:

AThe Sieve of Eratosthenes is used to find large prime numbers; its results will be inconsistent when used on small numbers, such as for the numbers less than 100.
BThe Sieve of Eratosthenes is meant to find prime numbers; it will not identify composite numbers.
CThe next prime number after 7 is 11. The smallest composite number which does not have 2, 3, 5, or 7 as a factor is 11², which is greater than 100.
DThe product 2 · 3 · 5 · 7 is greater than 100, so the Sieve will have crossed out all of the composite numbers less than 100.

Verified Answer :

  • A number \(n < 100\) is composite if it has a prime factor \(p \le \sqrt{n} \le 10\).
  • Primes up to 10 are 2, 3, 5, and 7.
  • Crossing out all multiples of these primes eliminates every composite number less than 100.
  • The next prime, 11, produces a smallest composite product of \(11^2 = 121 > 100\), so no other composite numbers remain.
  • The correct choice is C.
  • Final answer: \(\boxed{\text{C}}\)

Explanation

  1. Understanding Composite Numbers and Prime Factors

The Sieve of Eratosthenes is a classic algorithm used to identify all prime numbers up to a given limit, which in this case is 100. A number \(n\) is composite if it can be expressed as the product of two smaller natural numbers, which implies it must have at least one prime factor \(p\) such that \(p \le \sqrt{n}\).

  1. Applying the Criterion to Numbers Less Than 100

To determine if a number \(n < 100\) is composite, we check for prime factors \(p \le \sqrt{100}\). Since \(\sqrt{100} = 10\), any composite number less than 100 must have a prime factor that is less than or equal to 10. The prime numbers satisfying this condition are 2, 3, 5, and 7. Thus, any composite number less than 100 must be divisible by at least one of these primes.

  1. Conclusion and Identifying the Correct Option

By systematically crossing out the multiples of 2, 3, 5, and 7, we effectively eliminate all composite numbers less than 100. The next prime number after 7 is 11. Any composite number that does not have 2, 3, 5, or 7 as a factor must be formed by prime factors greater than or equal to 11. The smallest such composite number is \(11^2 = 121\). Since \(121 > 100\), there are no composite numbers less than 100 that remain after we have removed all multiples of 2, 3, 5, and 7. Therefore, the numbers remaining are prime. This reasoning confirms that option C is the correct explanation.

Examples

The Sieve of Eratosthenes is a fundamental tool for mathematicians and computer scientists working with prime numbers. Imagine you are working on a data security project and need to generate a list of all primes up to 1000 to use for cryptographic key generation. By applying the same logic—identifying all primes \(p\) such that \(p \le \sqrt{1000} \approx 31.6\)—you can efficiently clear all composite numbers by only crossing out multiples of primes up to 31. This method turns a potentially tedious task into a quick and structured process, illustrating how number theory concepts optimize real-world computational efficiency.

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