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The product of two prime numbers is $1994$, what is their sum.

So if you look at the factors of $1994$, you have: $1,2,997,1994$. Since $1$ is not considered prime. then the two numbers are $2$ and $997$, so their sum is $999$.

But is there a way to the find the two products ($2$ and $997$) without trial and error?

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  • $\begingroup$ For short: no. To factor an integer is generally quite a hard task. However, to test primality is way easier, so if we notice that $1994$ is an even number and test $1994/2$ for primality with an affirmative outcome, we are done. $\endgroup$ – Jack D'Aurizio Sep 26 '15 at 22:40
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Since $1994$ is even, and $2$ is the only even prime, we know immediately that one of the two prime factors is $2$. There is no choice but for the other prime to be $1994/2 = 997$.

Note that this is year-specific. For instance, one cannot ask the same question about $1996$ since $998$ is not a prime.

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