Properties of Numbers Divisible by 3

Oct 13, 2025 by miodeya.com 75 views

Introduction

In mathematics, the concept of divisibility is a fundamental aspect of number theory. It deals with the ability of a number to be divided by another number without leaving a remainder. In this article, we will explore the concept of divisibility by 3 and examine whether there are any numbers that are divisible by 3.

What is Divisibility by 3?

Divisibility by 3 is a property of numbers that can be divided by 3 without leaving a remainder. In other words, a number is divisible by 3 if it can be expressed as the product of 3 and another integer. For example, the numbers 3, 6, 9, 12, and 15 are all divisible by 3 because they can be expressed as 3 × 1, 3 × 2, 3 × 3, 3 × 4, and 3 × 5, respectively.

Properties of Numbers Divisible by 3

Numbers that are divisible by 3 have several interesting properties. One of the most notable properties is that the sum of the digits of a number that is divisible by 3 is also divisible by 3. For example, the number 123 is divisible by 3 because the sum of its digits (1 + 2 + 3 = 6) is also divisible by 3.

Examples of Numbers Divisible by 3

There are many examples of numbers that are divisible by 3. Some of the most common examples include:

Multiples of 3: Any multiple of 3 is divisible by 3. For example, 3, 6, 9, 12, 15, and so on.
Numbers with a sum of digits divisible by 3: As mentioned earlier, the sum of the digits of a number that is divisible by 3 is also divisible by 3. For example, 123, 456, 789, and so on.
Numbers that can be expressed as 3 × n: Any number that can be expressed as 3 × n is divisible by 3. For example, 3 × 1 = 3, 3 × 2 = 6, 3 × 3 = 9, and so on.

Are There Any Numbers That Are Not Divisible by 3?

Yes, there are many numbers that are not divisible by 3. Some examples include:

Numbers that leave a remainder when divided by 3: Any number that leaves a remainder when divided by 3 is not divisible by 3. For example, 1, 2, 4, 5, and so on.
Numbers that cannot be expressed as 3 × n: Any number that cannot be expressed as 3 × n is not divisible by 3. For example, 2, 4, 5, and so on.

Conclusion

In conclusion, there are many numbers that are divisible by 3. These numbers have several interesting properties, including the fact that the sum of their digits is also divisible by 3. However, there are also many numbers that are not divisible by 3. Understanding the concept of divisibility by 3 is an important aspect of number theory and has many practical applications in mathematics and other fields.

Frequently Asked Questions

What is the smallest number that is divisible by 3? The smallest number that is divisible by 3 is 3 itself.
What is the largest number that is divisible by 3? There is no largest number that is divisible by 3, as there are infinitely many multiples of 3.
How can I determine if a number is divisible by 3? You can determine if a number is divisible by 3 by checking if the sum of its digits is also divisible by 3.

References

"Number Theory" by G.H. Hardy and E.M. Wright
"Divisibility by 3" by Math Open Reference
"Properties of Numbers Divisible by 3" by Wolfram MathWorld

Glossary

Divisibility: The ability of a number to be divided by another number without leaving a remainder.
Multiple: A number that is the product of a given number and an integer.
Sum of digits: The sum of the individual digits of a number.

Introduction

In our previous article, we explored the concept of divisibility by 3 and examined the properties of numbers that are divisible by 3. In this article, we will answer some of the most frequently asked questions about divisibility by 3.

Q: What is the smallest number that is divisible by 3?

A: The smallest number that is divisible by 3 is 3 itself. This is because 3 is the smallest multiple of 3, and any number that is a multiple of 3 is divisible by 3.

Q: What is the largest number that is divisible by 3?

A: There is no largest number that is divisible by 3, as there are infinitely many multiples of 3. For example, 3, 6, 9, 12, 15, and so on are all divisible by 3.

Q: How can I determine if a number is divisible by 3?

A: You can determine if a number is divisible by 3 by checking if the sum of its digits is also divisible by 3. For example, the number 123 is divisible by 3 because the sum of its digits (1 + 2 + 3 = 6) is also divisible by 3.

Q: What are some examples of numbers that are not divisible by 3?

A: Some examples of numbers that are not divisible by 3 include:

Numbers that leave a remainder when divided by 3: Any number that leaves a remainder when divided by 3 is not divisible by 3. For example, 1, 2, 4, 5, and so on.
Numbers that cannot be expressed as 3 × n: Any number that cannot be expressed as 3 × n is not divisible by 3. For example, 2, 4, 5, and so on.

Q: Can I use a calculator to determine if a number is divisible by 3?

A: Yes, you can use a calculator to determine if a number is divisible by 3. Simply divide the number by 3 and check if the result is a whole number. If it is, then the number is divisible by 3.

Q: Are there any numbers that are divisible by 3 but not by 2?

A: Yes, there are many numbers that are divisible by 3 but not by 2. For example, 3, 9, 15, and so on are all divisible by 3 but not by 2.

Q: Are there any numbers that are divisible by 3 but not by 5?

A: Yes, there are many numbers that are divisible by 3 but not by 5. For example, 3, 9, 12, and so on are all divisible by 3 but not by 5.

Q: Can I use a formula to determine if a number is divisible by 3?

A: Yes, you can use a formula to determine if a number is divisible by 3. The formula is:

n = 3k

where n is the number and k is an integer.

Q: Are there any numbers that are divisible by 3 but not by 7?

A: Yes, there are many numbers that are divisible by 3 but not by 7. For example, 3, 9, 12, and so on are all divisible by 3 but not by 7.

Conclusion

In conclusion, we have answered some of the most frequently asked questions about divisibility by 3. We hope that this article has been helpful in understanding the concept of divisibility by 3 and how to determine if a number is divisible by 3.

Frequently Asked Questions

What is the smallest number that is divisible by 3? The smallest number that is divisible by 3 is 3 itself.
What is the largest number that is divisible by 3? There is no largest number that is divisible by 3, as there are infinitely many multiples of 3.
How can I determine if a number is divisible by 3? You can determine if a number is divisible by 3 by checking if the sum of its digits is also divisible by 3.

References

"Number Theory" by G.H. Hardy and E.M. Wright
"Divisibility by 3" by Math Open Reference
"Properties of Numbers Divisible by 3" by Wolfram MathWorld

Glossary

Divisibility: The ability of a number to be divided by another number without leaving a remainder.
Multiple: A number that is the product of a given number and an integer.
Sum of digits: The sum of the individual digits of a number.