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I need to find the cardinality of the set of all prime factors of $120$.

Will it be $16$? Since the set of all prime factors of $120$ will always have $16$ elements i.e $\{ 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120\}$.

Or is this a trick question and requires a proof to answer?

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1 Answer

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Definition: A integer $p>1$ is a prime number if it unique divisors are $p$ and $1$.

Fundamental theorem of arithmetic :

Every integer $n>1$ either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.

Solution:

With the definition above and the theorem, we can always find the prime factors of any integer $n$ (by fundamental theorem of arithmetic) .

Notice that we can write $$120= 2^3\times 3 \times 5$$ and Hence the prime factors are the set $$S=\lbrace 2,3,5\rbrace$$ as well.

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