Verified Answer :

  • Define a prime number as a natural number greater than 1 with no divisors other than 1 and itself.
  • Test 41 for prime factors using trial division up to \(\sqrt{41} \approx 6.4\); since it is not divisible by 2, 3, or 5, it is prime.
  • Observe that 45 is divisible by 5, 46 is divisible by 2, and 49 is divisible by 7, making them composite.
  • Conclude that the prime number is \(\boxed{41}\).

Explanation

  1. Definition and Analysis

The problem requires us to identify which of the given numbers—41, 45, 46, or 49—is a prime number. A prime number is defined as a natural number greater than 1 that has no positive divisors other than 1 and itself. To test if a number \(n\) is prime, we can check for divisibility by prime numbers less than or equal to \(\sqrt{n}\).

  1. Testing 41

We analyze each candidate number between 40 and 50 to determine its status:

  • For 41: The square root is \(\sqrt{41} \approx 6.4\). We check divisibility by prime numbers 2, 3, and 5.
  • 41 is not divisible by 2 (it is odd).
  • 41 is not divisible by 3 (the sum of digits \(4+1=5\) is not divisible by 3).
  • 41 is not divisible by 5 (it does not end in 0 or 5).

Since it is not divisible by any prime up to its square root, 41 is a prime number.

  1. Testing 45, 46, and 49

Now we check the remaining numbers:

  • 45: The sum of digits \(4+5=9\) is divisible by 3, and it ends in 5, so it is divisible by 3, 5, 9, and 15. Thus, 45 is composite.
  • 46: It is an even number ending in 6, so it is divisible by 2 (\(2 \times 23 = 46\)). Thus, 46 is composite.
  • 49: It is the square of 7 (\(7 \times 7 = 49\)), so it is divisible by 7. Thus, 49 is composite.
  1. Conclusion

Having tested all options, we confirm that 41 is the only prime number in the given set. The correct choice is 41. \(\boxed{41}\)

Examples

Identifying prime numbers is fundamental in fields like cryptography. For example, when you securely browse the internet (like accessing a bank website), your browser and the server use large prime numbers to create an encryption key that keeps your data safe. Understanding small primes like 41 helps as a foundation for learning how these security systems protect our sensitive information every day.

Related Questions