Question

State all prime numbers between 50 and 100 . \(\begin{array}{ll}\text { (a) } 30 & \text { (b) } 96\end{array}\) (c) 500 (d) 589 (e) 3239

Short answer

(a) 30 (b) 96 (c) 500 (d) 589 (e) 3239 Answer: None of them

Step-by-step solution

Verify if a number is prime

For a number to be prime, it must have only two factors: 1 and itself. Divide by every number less than the square root of the number in question, and if none of them divides the number without a remainder, then it is a prime number.

Find prime numbers between 50 and 100

We'll check each number in the range 50 to 100 one by one and verify if it is prime. Remember that 1 and 0 are not prime numbers, and all even numbers except 2 are not prime since they can be divided by 2. We can now list down prime numbers between 50 and 100: 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Cross-check with given options

Now, we compare the prime numbers we found with the given options: (a) 30 - Not in the range we found. (b) 96 - Not in the range we found. (c) 500 - Not in the range we found. (d) 589 - Not in the range we found. (e) 3239 - Not in the range we found. Since none of the options is a prime number between 50 and 100, the correct answer is "none of them".

Key concepts

Prime Number

A prime number is a unique type of whole number that has precisely two distinct positive divisors: 1 and itself. This means that a prime number cannot be divided evenly by any other numbers apart from 1 and its own value. For instance, the number 7 is prime because the only ways to divide it are 1 x 7 or 7 x 1, with no other division giving a whole number result.

When searching for prime numbers between 50 and 100, one must individually test each number in that range to see if it adheres to the prime definition. Sometimes this process can be simplified using the concept of the square root, which can reduce the number of tests required for larger numbers, as you'll see in the 'Square Root' section.

Factors of Numbers

Factors of a number are the whole numbers that can divide that number without leaving a remainder. In other words, if you can multiply two whole numbers to get the original number, those two are considered its factors. For example, the factors of 10 are 1, 2, 5, and 10 because 1 x 10 = 10 and 2 x 5 = 10.

Recognizing factors is essential in determining whether a number is prime. A prime number should only have two factors, as mentioned previously. However, while seeking prime numbers between 50 and 100, they cannot have any other factors besides 1 and themselves. Thus, understanding factors helps in quickly eliminating non-prime candidates by identifying those that have additional divisors.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 x 5 equals 25.

Using the square root in the context of identifying prime numbers between 50 and 100 is very useful because you only need to check for factors up to the square root of the number in question. This is because any factor greater than the square root would need to be multiplied by a number less than the square root to reach the target number, and you would have already found this smaller factor during the earlier checks. Therefore, the square root simplifies the process of prime identification by reducing the number of potential factor candidates, thereby expediting the search within a range such as between 50 and 100.