Question

How many divisors do the numbers 23,41 and 79 have (do not count the same divisor more than once; for example, all are divisible by 1 but only count 1 once in your answer)?

Short answer

Each number has 2 divisors.

Step-by-step solution

Identify Prime Numbers

First, determine whether each number (23, 41, and 79) is a prime number. A prime number has exactly two distinct positive divisors: 1 and itself.

Verify the Primality

Check if each number (23, 41, and 79) is divisible by any integer other than 1 and itself. If they are not, they are prime numbers.

List the Divisors of Prime Numbers

Since 23, 41, and 79 are prime, their divisors are just 1 and the number itself.

Count the Divisors

Each prime number 23, 41, and 79 has exactly 2 divisors.

Key concepts

Understanding Divisors

Divisors are the numbers that divide a given number exactly, without leaving any remainder. In simple terms, a divisor is a number that fits perfectly into another. For example, for the number 6, its divisors are 1, 2, 3, and 6. If you divide 6 by any of these numbers, you get a whole number with no remainder.

It's helpful to consider small numbers first to get a grasp of divisors. Let's take 10 as an example. By dividing 10 by 1, 2, 5, and 10, we get whole numbers: ewline 10 / 1 = 10 ewline 10 / 2 = 5 ewline 10 / 5 = 2 ewline 10 / 10 = 1

Hence, 1, 2, 5, and 10 are the divisors of 10. Notice that 3 and 4 are not because they produce remainders when divided into 10.
Understanding this basic concept will help you better navigate more complex topics in math, such as prime numbers and factoring.

Identifying Prime Numbers

A prime number is a special type of number that has exactly two divisors: 1 and the number itself. This means it cannot be divided evenly by any other numbers. For example, let's test if 23 is a prime number.

To identify if a number is prime:

• First, ignore 1 and the number itself.
• Next, check if it can be divided by any other numbers up to its square root (for 23, this would be approximately 4.8).

If none of these divisions result in a whole number, it is confirmed to be prime. For 23: ewline 23 / 2 ≈ 11.5 ewline 23 / 3 ≈ 7.67 ewline 23 / 4 ≈ 5.75

Since none of these calculations resulted in whole numbers, 23 is prime. Applying the same steps to 41 and 79 shows they are also prime because they only have two divisors each (1 and themselves).

Remember, understanding how to identify prime numbers is a powerful tool in math problem solving.

Math Problem Solving

Approaching math problems methodically can make even the toughest questions manageable. Let's walk through a systematic method to solve this divisors problem:

1. **Understand the Problem**: We're asked to find how many divisors the numbers 23, 41, and 79 have.

2. **Break It Down**: Identify whether these numbers are prime, by checking if each has any divisors other than 1 and itself.

3. **Apply Knowledge**: Since prime numbers only have two divisors (1 and itself), and we determined that 23, 41, and 79 are all prime, we can state each has exactly two divisors.

4. **Cross-Verify**: Double-check our calculations to confirm no other numbers divide 23, 41, and 79 evenly apart from 1 and themselves.

Following these steps helps ensure accuracy and builds confidence in problem-solving abilities. Always break down complex problems into small, manageable steps. This strategy is essential to mastering math challenges.