Let p and q be distinct positive prime numbers such that \(p > q\) and \(p^2 - q^2 = 45\). What is the value of \(p + q\)?
9
We know \(p^2 - q^2 = (p-q)(p+q) = 45\).
If both p and q were odd primes, \(p-q\) would be even, but 45 is odd, so this is not possible.
Hence one of the primes must be even, so \(q = 2\) (the only even prime).
Then \(p^2 - 4 = 45 \Rightarrow p^2 = 49 \Rightarrow p = 7\), which is indeed prime and greater than q.
So \(p + q = 7 + 2 = 9\).
Hence, the value of \(p+q\) is 9.
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