Question

What is the value of the integer \(p ?\) (1) \(p\) is a prime number. (2) \(88 \leq p \leq 95\)

Short answer

The value of p is 89.

Step-by-step solution

Understand the Range

According to the second statement, the integer p must lie within the range: \[88 \leq p \leq 95\].

Identify Prime Numbers in the Range

List all prime numbers between 88 and 95. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. Therefore, examine each number in the range to see if it's prime: 88 (not prime), 89 (prime), 90 (not prime), 91 (not prime), 92 (not prime), 93 (not prime), 94 (not prime), and 95 (not prime).

Determine the Value of p

There is only one prime number in the range: 89. Therefore, the value of p must be 89.

Key concepts

Understanding Prime Numbers

Prime numbers are foundational in number theory and crucial for many mathematical applications. A prime number is an integer greater than 1 that has no positive divisors other than 1 and itself. This means that if you try to divide a prime number by any other number except 1 and itself, you will get a remainder. For instance, 7 is a prime number because the only divisors of 7 are 1 and 7. Conversely, 6 is not a prime number because it can be divided evenly by 1, 2, 3, and 6.
To identify a prime number, follow these steps:

• Check divisibility by numbers starting from 2 up to the square root of the number in question.
• If none of these numbers divide the number entirely (i.e., without leaving a remainder), then the number is prime.

Understanding primes is central to this exercise as it helps narrow down possibilities in a given number range.

Exploring Integer Properties

Integer properties help us understand the characteristics and behavior of whole numbers. In this exercise, focus on the following essential integer properties:

• Divisibility: An integer is divisible by another if dividing them yields a whole number. For example, 15 is divisible by 3 because 15 ÷ 3 = 5.
• Even and Odd Numbers: Even numbers are divisible by 2 (e.g., 2, 4, 6). Odd numbers are not divisible by 2 (e.g., 1, 3, 5).
• Prime vs. Composite Numbers: Prime numbers have exactly two distinct positive divisors: 1 and themselves. Composite numbers have more than two divisors.

These properties are useful when evaluating and categorizing numbers within a specified range, as we did in the exercise to identify potential values for p.

Analyzing the Number Range

In numerical problems, analyzing the given number range simplifies problem-solving. For our exercise, the range is 88 ≤ p ≤ 95. Understanding what this range means and the best ways to approach it can be incredibly insightful:

• Identify the boundaries: Recognize the lowest and highest values in the range. Here, it’s 88 and 95.
• Check each number within the range: Look at every number between the lower and upper bounds and evaluate them based on the problem's requirements. For prime number identification, you do this by checking each number for primality.
• Eliminate non-qualifying numbers: Use properties like divisibility to swiftly exclude numbers that do not meet the specifications (e.g., non-prime numbers).

We narrow down the value of p by checking if each number from 88 to 95 is a prime number, concluding that only 89 fits. This systematic approach highlights the efficiency of breaking down the number range and leveraging integer properties.