Problem 4 - Discrete Mathematics with Applications

Question

Determine if each positive integer is a prime. $$1723$$

Step-by-Step Solution

Answer

We were given the number \(1723\) to check if it is prime. After calculating its square root (approximately \(41.5\)), we checked for factors from \(2\) to \(41\). None of these numbers evenly divided \(1723\), revealing that it only has two unique factors: 1 and itself. Thus, \(1723\) is a prime number.

Step 1: Identify the number

We have been given the number \[1723\] to determine if it is a prime number.

Step 2: Calculate the square root

Calculate the square root of the given number, since we only need to check for factors up to the square root of the number. In this case, the square root of \(1723\) is approximately \(41.5\). We can round the value down to 41 since we only need to consider integer factors.

Step 3: Check for factors

Now we will check for any factors of \(1723\) in the range of \(2\) to \(41\), inclusive. Determine if any of these numbers can evenly divide \(1723\).

Step 4: Analyze the results

After checking, we find that none of the numbers from \(2\) to \(41\) can evenly divide \(1723\). Therefore, \(1723\) has only two unique factors: 1 and itself (1723).

Step 5: Determine if the number is prime

Since \(1723\) has only two unique factors (1 and itself) and no other factors were found in the given range, we can conclude that \(1723\) is a prime number.

Key Concepts

Integer Factorization

Factorization is the process of breaking down a number into its divisors, or factors, which when multiplied together produce the original number. For large numbers, checking each possible factor can be time-consuming, so we refine the process by only checking up to the square root of the number. If a large number can be divided evenly by other numbers, those numbers are its factors.

• An integer factor is a whole number that divides another number exactly, leaving no remainder.
• To find if a number is composite, we search for factors other than 1 and itself.
• If no such factors exist, the number is prime.

Understanding integer factorization is critical for evaluating whether numbers like 1723 are prime.

Square Root Estimation

Estimating the square root of a number helps narrow down the range of numbers to test for factors. For our example with 1723, calculating the square root gives us about 41.5. We round down to 41 because we only check for integer factors. Why check up to only the square root?

• Any factor larger than the square root would have a corresponding smaller factor.
• This effectively reduces the workload, making factorization faster and more efficient.

This method saves time and effort, simplifying the process of identifying prime numbers.

Divisibility

Divisibility is key to determining prime numbers. A prime number is only divisible by 1 and itself. To check divisibility,

• Start with the smallest prime, 2, and proceed upwards.
• Use simple rules like even numbers being divisible by 2.

In our example, 1723 is tested against all numbers up to 41 for divisibility. If any number divides 1723 without a remainder, it would be a factor. Being meticulous here ensures correct identification of prime numbers.

Unique Factors

Every number has factors, but prime numbers are special because they have exactly two. The number 1 and the number itself form a unique pair of factors that no other numbers share.

• Checking for unique factors reveals if a number is prime or composite.
• Prime numbers, like 1723, can't be factored further beyond its own unique factors of 1 and itself.

This uniqueness of factors categorizes numbers as prime, which is essential for understanding their properties and applications in mathematics.