Question

The number of multiples of a given number is finite.
(A) True
(B) False

Answer 1

Hint: The question requires you to check whether the given statement is true or false. Assume a given number as ‘m’ and multiply natural numbers to it, i.e. $m \times 1,m \times 2,m \times 3,m \times 4,......$, which represents the multiples of the given number. Now use this conclusion to find the correct option among the two.

Complete step-by-step answer:
Here in this problem, we are given a statement “the number of multiples of a given number is finite”. And we need to find out whether the given statement is true or false.
Before starting with the solution of this problem, we should understand the concept of multiples. A multiple of a number is a number that is the product of a given number and some other natural number. Multiples can be observed in a multiplication table.
Multiples of $2$ are $2,4,6,8,10,12,14{\text{ }}$ and so on.
Multiples of $3$ are $3,6,9,12,15,18,21$ and so on.
Multiples of $5$ are $5,10,15,20,25,30,35,40$ and so on.
For example, in this case $2 \times 5 = 10$ , $2$ and $5$ are multiplied to get the product as $10$. Here $10$ is the multiple of both $2$ and $5$ . And $2$ & $5$ are factors of $10$ .
Let us assume that a given number is $m$ .
So according to the definition of multiples of a number, any natural number multiplied with the given number will result in a different multiple of that number, i.e.
$ \Rightarrow m \times 1,m \times 2,m \times 3,m \times 4,......$are all representing a multiple of number ‘m’.
Since we know that there are infinitely many natural numbers, we can conclude that there can be infinitely many multiples of a given number.
Hence, we can say that option (B) is correct, which implies that the given statement “the number of multiples of a given number is finite” is a false statement.

Note: In questions like this the understanding of the definition of terms likes multiples and factors play a crucial role. An alternative approach for the same problem can be to assume the given statement as true at the beginning of the solution. Then use an example to check whether the assumption is correct or not. If the example contradicts your assumption, another option will be the correct answer. If it doesn’t contradict, then the assumed option will be the correct one.