Problem 14 - Algebra A Combined Function

Question

Identify each number as prime or composite. See Example \(3 .\) 53

Step-by-Step Solution

Answer

53 is a prime number.

Step 1: Understand Prime and Composite Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. On the other hand, a composite number has additional divisors apart from 1 and itself.

Step 2: Consider Divisibility Rules

To determine if 53 is composite, check if it is divisible by any prime number up to the square root of 53, which is approximately 7.28.

Step 3: Check Divisibility by 2

53 is not divisible by 2 because it is an odd number.

Step 4: Check Divisibility by 3

To check divisibility by 3, add the digits of 53 (5 + 3 = 8) and check if the result is divisible by 3. Since 8 is not divisible by 3, 53 is not divisible by 3.

Step 5: Check Divisibility by 5

A number is divisible by 5 if it ends in 0 or 5. 53 ends in 3, so it is not divisible by 5.

Step 6: Check Divisibility by Other Numbers

The next prime number to check is 7. Since 53 is not divisible by 7 (53 divided by 7 is approximately 7.57, not an integer), 53 is not divisible by any numbers up to its square root.

Step 7: Conclusion

Since 53 is not divisible by any prime numbers up to its square root, it is a prime number.

Key Concepts

Divisibility Rules

Divisibility rules are simple shortcuts to help you figure out whether one number divides evenly into another without actually performing the division. These rules can save a lot of time and effort, especially when determining if a number is prime or composite. Here are some handy divisibility checks you can use:

• Divisibility by 2: A number is divisible by 2 if it's an even number. This means its last digit must be 0, 2, 4, 6, or 8.
• Divisibility by 3: Add all the digits in the number together. If the resulting sum is a multiple of 3, then the original number is divisible by 3.
• Divisibility by 5: If a number ends in 0 or 5, it is divisible by 5.
• Divisibility by 7: For this test, double the last digit and subtract it from the rest of the number. If the result is divisible by 7, then the original number is too.

These rules help in identifying whether a number has divisors other than 1 and itself, hence playing a crucial role in determining if a number is prime or composite.

Prime Numbers

Prime numbers are the building blocks of all numbers. A prime number is defined as a natural number greater than 1 that can only be divided by 1 and itself without leaving a remainder.

• Only Two Divisors: The uniqueness of prime numbers lies in their having exactly two divisors.
• Examples of small prime numbers: 2, 3, 5, 7, 11, and 13. Note that 2 is the only even prime number.

Understanding prime numbers is vital because they are the simplest form of numbers and are used in various mathematical applications, including cryptography and number theory.
Whenever determining if a number is prime, always check dividing by smaller prime numbers up to the square root of the number of interest. If none divide it, the number is prime.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted as \( \sqrt{n} \), where \( n \) is the number whose square root you're finding.

• Why Use the Square Root: When testing if a number is prime, only prime numbers up to its square root need to be checked for divisibility. This is because factors, if they exist, appear as pairs. If no factor is found below the square root, none will exist above it, making the square root a valuable tool for efficiency.
• Example of Use: To check if 53 is a prime number, calculate the square root. Since \( \sqrt{53} \) is approximately 7.28, check divisibility using prime numbers up to 7, such as 2, 3, 5, and 7.

Working with square roots is an essential technique in simplifying the process of identifying prime numbers, saving time and effort.