Answered

The numbers 2, 3, 5, 7, 11, 13, and 17 are not factors of 293. Explain why it is possible to conclude that 293 is prime without checking for more prime factors.



Choose the correct answer below.


AThe product of the three highest primes in the list, 11, 13, and 17, is greater than 293, so if 293 does not have a common factor with any of these primes, then it is also prime.
BThe product of the primes listed has no common factors with 293, so 293 must be prime.
CThe square of 17 is 289 and the square of the next prime, 19, is 361, so 293 is not divisible by any prime whose square is less than 293.
DThe square of 17 is 289 and there are no prime numbers between 289 and 293, so 293 must also be prime.

Verified Answer :

  • To check if a number \(n\) is prime, test all prime factors up to \(\lfloor \sqrt{n} \rfloor\).
  • For \(n=293\), we find \(\lfloor \sqrt{293} \rfloor = 17\) because \(17^2=289 < 293 < 19^2=361\).
  • Since none of the primes \(\{2, 3, 5, 7, 11, 13, 17\}\) divide \(293\), it must be prime.
  • The correct choice is C. \(\boxed{C}\)

Explanation

  1. The Primality Test Principle

To determine if an integer \(n\) is prime, we only need to test for prime factors \(p\) such that \(p \leq \sqrt{n}\). If a number \(n\) is composite, it must have at least one prime factor \(p\) such that \(p \leq \sqrt{n}\). Otherwise, if no such prime factor exists, the number \(n\) must be prime. In this case, we are checking the number \(n = 293\).

  1. Determining the Required Primes to Test

To find the bound for testing, we calculate \(\sqrt{293}\). Since \(17^2 = 289\) and \(18^2 = 324\), we know that \(17 < \sqrt{293} < 18\). Thus, \(\lfloor \sqrt{293} \rfloor = 17\). The prime numbers we need to test are all primes less than or equal to \(17\), which are \(2, 3, 5, 7, 11, 13, \text{ and } 17\). Since we are given that none of these primes are factors of \(293\), we can conclude that \(293\) has no prime factors up to \(\sqrt{293}\).

  1. Evaluating the Options

Now we evaluate the provided options:

AThe product of the three highest primes is not a standard test for primality.
BThis does not follow the mathematical criterion for testing primality.
CThis option states that the square of \(17\) is \(289\) and the square of the next prime, \(19\), is \(361\), implying that \(293\) is not divisible by any prime whose square is less than or equal to \(293\). This aligns perfectly with the requirement to test primes up to \(\sqrt{293}\).
DThe presence or absence of prime numbers between \(289\) and \(293\) is irrelevant to the definition of primality for \(293\).

Therefore, option C provides the correct reasoning.

  1. Final Conclusion

The fact that \(293\) is not divisible by any prime \(p \le \sqrt{293} = 17\) confirms its primality, as any composite number must have a prime factor not exceeding its square root. Since \(17^2 = 289 < 293\) and the next prime \(19^2 = 361 > 293\), testing up to \(17\) is sufficient. Thus, the correct conclusion is C. \(\boxed{C}\)

Examples

In cryptography, primality testing is essential for generating keys. When an algorithm needs to ensure a large number is prime, it doesn't need to check all numbers up to the value itself; instead, it only checks prime factors up to the square root of that number. For instance, if you were verifying if 293 is a suitable "prime" for a simplified encryption task, realizing you only need to test up to \(\lfloor \sqrt{293} \rfloor = 17\) saves significant computational time, making security protocols efficient and fast.

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