Problem 4 - Elementary and Intermediate Algebra

Question

In each of Exercises \(1-4\) match the description with the appropriate number from the list on the right. ___The only even prime number A. 2 B. 7 C. 60 D. 65

Step-by-Step Solution

Answer

A. 2

Step 1: Understand Prime Numbers

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. For example, 2, 3, 5, and 7 are prime numbers.

Step 2: Identify Even and Odd Numbers

A number is even if it is divisible by 2. In other words, an even number can be expressed as 2n where n is an integer. In the given list, only 2 and 60 are even numbers.

Step 3: Determine Prime Numbers in the Options

From the list provided, 2 is a prime number because it has no divisors other than 1 and itself. Check if the other even number (60) is prime: 60 has divisors other than 1 and itself (e.g., 2, 3, 5, etc.), so it is not prime.

Step 4: Conclude the Answer

Since the only even prime number is 2, and it is the only number in the list that satisfies this condition, the answer is A. 2.

Key Concepts

Even Numbers

Even numbers are integers that can be exactly divided by 2. This means that they have no remainder when divided by 2. For an easier understanding, any number of the form 2n, where n is an integer, is considered even.

Some characteristics of even numbers include:

• The last digit in an even number is always 0, 2, 4, 6, or 8.
• When you add, subtract, or multiply two even numbers, the result is still an even number.
• Only one of the even numbers can be a prime, which is 2.

Using the given list of numbers, which are 2, 7, 60, and 65, you can easily identify 2 and 60 as even numbers because they are divisible by 2.

Divisors

Divisors are numbers that divide another number without leaving a remainder. For instance, if we consider the number 28, its divisors are 1, 2, 4, 7, 14, and 28.

Here’s how to find the divisors of a number:

• Start with 1 and the number itself, as these are always divisors.
• Check each integer less than the given number to see if it divides the number without a remainder.

In the context of the exercise, we need to determine if numbers like 2 and 60 are prime by examining their divisors. The number 2 has only two divisors, 1 and 2, making it a prime. In contrast, 60 has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60) and thus, it is not a prime number.

Number Theory

Number theory involves the study of natural numbers or integers, and it explores their properties, patterns, and relationships. It focuses heavily on understanding prime numbers and their significance in mathematics.

Prime numbers are special because they only have two divisors: 1 and the number itself. In the exercise, correctly identifying the prime numbers is crucial because only one of the even numbers can be a prime number.

• Prime numbers greater than 2 are always odd, but 2 itself is the only even prime number.
• Prime numbers are fundamental in number theory because they act like the 'building blocks' of the integers.

Understanding number theory helps students grasp why 2 is unique as the only even prime number, which is a key point in arriving at the solution of the exercise.