Problem 19 - Algebra A Combined Function

Question

Identify each number as prime or composite. See Example \(3 .\) 2065

Step-by-Step Solution

Answer

2065 is a composite number.

Step 1: Understand Prime and Composite Numbers

A prime number has only two distinct positive divisors: 1 and itself. A composite number has more than two divisors. The task is to determine whether 2065 is prime or composite.

Step 2: Check Divisibility by 2

The number 2065 is odd, meaning it is not divisible by 2. Therefore, we need to check other potential factors.

Step 3: Check Divisibility by 3

To check if a number is divisible by 3, sum its digits and see if the result is divisible by 3. The digits of 2065 sum up to 2 + 0 + 6 + 5 = 13, which is not divisible by 3. Thus, 2065 is not divisible by 3.

Step 4: Check Divisibility by 5

A number is divisible by 5 if it ends in 0 or 5. Since 2065 ends in 5, it is divisible by 5.

Step 5: Determine if 2065 is Composite

Since 2065 is divisible by 5 and has a divisor other than 1 and itself, it is a composite number.

Key Concepts

Divisibility Rules

Divisibility rules are simple tricks or methods that help quickly determine if one number is a divisor of another without performing full division. Understanding these rules can save you a lot of time, especially when deciding whether a number is prime or composite. Different divisibility rules apply to different numbers. For example:

• A number is divisible by 2 if it is even, meaning its last digit is 0, 2, 4, 6, or 8.
• To check for divisibility by 3, add up all the digits in the number. If the sum is divisible by 3, then the original number is also divisible by 3.
• A number can be divided by 5 if it ends in 0 or 5.

For the number 2065, the divisibility rules were used to determine that it is divisible by 5, since it ends with a 5. This rule helped us quickly identify that 5 is a divisor of 2065, leading to the conclusion that it is a composite number.

Prime Numbers

Prime numbers are the building blocks of all integers. A prime number is defined as a number greater than 1 that has no divisors other than 1 and itself. This means if you have a prime number, you cannot divide it evenly by any other integers besides 1 and the number itself.

For example, the number 3 is prime because it is not divisible by numbers other than 1 and 3. When assessing whether a number is prime, one commonly uses divisibility rules to see if others evenly divide it. If none of these numbers divide the original number evenly (except 1 and itself), then the number is prime.

Finding prime numbers becomes particularly useful because any composite number can be expressed as a product of prime numbers, a concept known as prime factorization.

Composite Numbers

Composite numbers are the opposite of prime numbers. They have more than two distinct positive divisors. In other words, besides being divisible by 1 and the number itself, they can be divided evenly by additional numbers, meaning they can be factored into smaller integers.

• An example is the number 4, which can be divided by 1, 2, and 4.
• The number 6 can be divided by 1, 2, 3, and 6.

To identify a composite number, you can use divisibility rules and see if any smaller integers divide into the number without leaving a remainder.

In the case of 2065, using the divisibility rule for 5 helped determine it is not just divisible by 1 and 2065, but also by 5. This additional divisor confirms that 2065 is a composite number. Identifying composite numbers helps in breaking down numbers into their prime factorized form, which is crucial for many math processes and problem-solving situations.