Yes, there are infinite multiples of any given number.
Understanding Multiples
A multiple of a number is the result of multiplying that number by an integer. For example, the multiples of 2 are 2, 4, 6, 8, and so on. We can easily generate multiples by repeatedly adding the original number to itself or by using the multiplication operation with positive integers.
Key Concepts:
- Integers: Numbers without fractions, including positive, negative, and zero. (e.g., -3, -2, -1, 0, 1, 2, 3)
- Multiplication: The arithmetic operation of combining groups of equal sizes.
The Infinity of Multiples
The reference material provided states: "Since we know that there are infinitely many natural numbers, we can conclude that there can be infinitely many multiples of a given number." This is because you can multiply any number by infinitely many natural numbers (1, 2, 3, ...), generating an infinite sequence of multiples.
Examples:
- Multiples of 3: 3, 6, 9, 12, 15, 18, 21... (and so on, infinitely)
- Multiples of 5: 5, 10, 15, 20, 25, 30, 35... (and so on, infinitely)
- Multiples of 10: 10, 20, 30, 40, 50, 60, 70... (and so on, infinitely)
Proving Infinity
The fact that there are infinitely many natural numbers directly leads to the conclusion that multiples are also infinite. Each new natural number used as a multiplier will create a new unique multiple of the starting number. There is no limit to how high a natural number can get, so there is no limit to the number of multiples.
Conclusion
In conclusion, the number of multiples of a given number is indeed infinite. This is derived from the infinite nature of natural numbers that are used in multiplication. The statement "the number of multiples of a given number is finite” is, therefore, a false statement.
