Write natural numbers between 50 and 65. What fraction of them are prime number
The correct answer and explanation is:
The natural numbers between 50 and 65 are:
51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64
To find what fraction of these numbers are prime, first identify which numbers are prime.
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
Check each number:
- 51: divisible by 3 and 17 → not prime
- 52: divisible by 2 → not prime
- 53: no divisors other than 1 and 53 → prime
- 54: divisible by 2 and 3 → not prime
- 55: divisible by 5 and 11 → not prime
- 56: divisible by 2 and 7 → not prime
- 57: divisible by 3 and 19 → not prime
- 58: divisible by 2 and 29 → not prime
- 59: no divisors other than 1 and 59 → prime
- 60: divisible by 2, 3, 5 → not prime
- 61: no divisors other than 1 and 61 → prime
- 62: divisible by 2 and 31 → not prime
- 63: divisible by 3 and 7 → not prime
- 64: divisible by 2 → not prime
Prime numbers in this range are 53, 59, and 61.
There are 14 natural numbers between 50 and 65.
Out of these, 3 are prime.
The fraction of prime numbers is therefore 3/14.
Explanation:
The problem involves two steps: listing natural numbers in a specified range and identifying which of those numbers are prime. Prime numbers are fundamental in number theory because they serve as the building blocks of natural numbers. To determine if a number is prime, the usual method involves checking divisibility by all integers greater than 1 and less than or equal to the square root of that number. For example, to check if 53 is prime, one only needs to test divisibility by prime numbers up to 7 (since 7² = 49 and 8² = 64, slightly above 53), such as 2, 3, 5, and 7. Since 53 is not divisible by any of these, it is prime. Applying this method systematically to the numbers between 50 and 65 identifies 53, 59, and 61 as primes. Finding the fraction as 3/14 gives a proportion of primes in this range. This fraction cannot be simplified further, so it represents the exact ratio of prime numbers to the total natural numbers between 50 and 65.