Question

Every counting number has an infinite number of ________?
A) Factors
B) Multiples
C) Prime factors
D) None of these

Answer 1

Hint:
Here we can consider multiples of a number by considering its products with natural numbers. We know that the set of natural numbers is infinite. Options A and C can be seen wrong using counter examples. Thus we get the answer.

Complete step by step solution:
Let $n$ be a counting number.
Then we can consider $2n, 3n, 4n, ....$
Choosing each natural number we get a multiple of the number $n$.
Since the set of natural numbers is infinite, we can say there are an infinite number of multiples of $n$.
This gives option B is right.
We can check other options too.
Option A is wrong.
A number need not have an infinite number of factors.
A number is said to be a factor of another number if the latter can be expressed as a product of the former and any other number.
For example, consider $5$.
Only factors of $5$ are $1$ and $5$. (since $5$ can be uniquely expressed as $1 \times 5$)
Similarly option C is also wrong.
Number of prime factors need not be infinite.
Consider $15$.
We know, $15 = 1 \times 15 = 3 \times 5$
So its factors are $1, 3, 5, 15$.
Among them prime factors are $3$ and $5$.

Therefore the answer is option B.

Note:
To prove a statement true we have to make a general proof. But to disprove something a counter example is enough. So, here we had shown that options A and C are wrong and also B is right.