# Product of Two Distinct Primes has 4 Positive Divisors

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## Theorem

Let $n \in \Z_{>0}$ be a positive integer which is the product of $2$ distinct primes.

Then $n$ has exactly $4$ positive divisors.

## Proof

Let $n = p \times q$ where $p$ and $q$ are primes.

We have by definition of divisor:

\(\displaystyle 1\) | \(\divides\) | \(\displaystyle n\) | One Divides all Integers | ||||||||||

\(\displaystyle p\) | \(\divides\) | \(\displaystyle n\) | Definition of Divisor of Integer | ||||||||||

\(\displaystyle q\) | \(\divides\) | \(\displaystyle n\) | Definition of Divisor of Integer | ||||||||||

\(\displaystyle p \times q\) | \(\divides\) | \(\displaystyle n\) | Integer Divides Itself |

where $\divides$ denotes divisibility.

Suppose $a \divides n$ such that $1 \le a < n$.

Suppose $a \ne p$.

By definition of prime number:

- $a \perp p$

where $\perp$ denotes coprimality.

From Euclid's Lemma:

- $a \divides q$

and so by definition of prime number:

- $a = q$

or:

- $a = 1$

Similarly, suppose $a \ne q$.

By the same reasoning:

- $a = p$

or:

- $a = 1$

Thus the only positive divisors are as above.

$\blacksquare$

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $33$