April 15, 2024
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Is 73 a Prime Number?

Is 73 a Prime Number?

 

Introduction

Prime numbers have always fascinated mathematicians and enthusiasts alike. They are the building blocks of the number system, possessing unique properties that make them intriguing. In this article, we will explore the question: Is 73 a prime number? We will delve into the definition of prime numbers, examine the divisibility rules, and provide evidence to determine whether 73 is indeed a prime number or not.

Understanding Prime Numbers

Before we dive into the specifics of 73, let’s first establish what prime numbers are. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In simpler terms, it is a number that cannot be evenly divided by any other number except 1 and itself.

Divisibility Rules

To determine whether a number is prime or not, we can apply various divisibility rules. These rules help us identify if a number can be divided evenly by another number without leaving a remainder. Let’s examine some of the common divisibility rules:

  • Divisible by 2: If the last digit of a number is even (0, 2, 4, 6, or 8), then the number is divisible by 2.
  • Divisible by 3: If the sum of the digits of a number is divisible by 3, then the number itself is divisible by 3.
  • Divisible by 5: If the last digit of a number is either 0 or 5, then the number is divisible by 5.
  • Divisible by 7: There is no simple rule for divisibility by 7, and it requires more complex calculations.

Is 73 a Prime Number?

Now, let’s apply the divisibility rules to determine whether 73 is a prime number or not.

Divisibility by 2

Since 73 ends with an odd digit (3), it is not divisible by 2. Therefore, we can conclude that 73 is not an even number.

Divisibility by 3

To check if 73 is divisible by 3, we sum its digits: 7 + 3 = 10. Since 10 is not divisible by 3, we can conclude that 73 is not divisible by 3.

Divisibility by 5

As 73 does not end with a 0 or 5, it is not divisible by 5. Therefore, we can conclude that 73 is not divisible by 5.

Divisibility by 7

Divisibility by 7 requires more complex calculations. However, we can use a simple method called “casting out sevens” to determine if a number is divisible by 7. We subtract twice the last digit from the remaining leading truncated number. If the result is divisible by 7, then the original number is also divisible by 7.

For 73, we subtract twice the last digit (3) from the remaining leading truncated number (7). This gives us 7 – (2 * 3) = 7 – 6 = 1. Since 1 is not divisible by 7, we can conclude that 73 is not divisible by 7.

Conclusion

After applying the divisibility rules, we can confidently state that 73 is a prime number. It is not divisible by 2, 3, 5, or 7, which are the common divisibility rules we explored. Therefore, 73 remains a prime number.

Key Takeaways

  • Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.
  • Divisibility rules help determine if a number is divisible by another number without leaving a remainder.
  • 73 is a prime number as it is not divisible by 2, 3, 5, or 7.

Q&A

1. What is a prime number?

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

2. What are some common divisibility rules?

Some common divisibility rules include divisibility by 2, 3, 5, and 7. For example, a number is divisible by 2 if its last digit is even.

3. How can we determine if a number is divisible by 7?

Divisibility by 7 requires more complex calculations. One method is to subtract twice the last digit from the remaining leading truncated number. If the result is divisible by 7, then the original number is also divisible by 7.

4. Is 73 divisible by 2?

No, 73 is not divisible by 2 as it ends with an odd digit (3).

5. Is 73 divisible by 3?

No, 73 is not divisible by 3 as the sum of its digits (7 + 3 = 10) is not divisible by 3.

6. Is 73 divisible by 5?

No, 73 is not divisible by 5 as it does not end with a 0 or 5.

7. Is 73 divisible by 7?

No, 73 is not divisible by 7 as the result of subtracting twice the last digit (3) from the remaining leading truncated number (7) is not divisible by 7.

8. Is 73 a prime number?

Yes, 73 is a prime number as it is not divisible by 2, 3, 5, or 7.

Summary

Prime numbers are fascinating mathematical entities, and determining whether a number is prime or not can be an intriguing exercise. In this article, we explored the question of whether 73 is a prime number. By applying the divisibility rules, we concluded that 73 is indeed a prime number. It is not divisible by 2, 3, 5, or 7, which are the common divisibility rules we examined. Understanding prime numbers and their properties can deepen our appreciation for the beauty and complexity of mathematics

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Diya Patel

Diya Patеl is an еxpеriеncеd tеch writеr and AI еagеr to focus on natural languagе procеssing and machinе lеarning. With a background in computational linguistics and machinе lеarning algorithms, Diya has contributеd to growing NLP applications.

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