If $$a, b,$$ and $$c$$ are three prime numbers such that $$abc = 23(a + b + c)$$, then the maximum possible value of $$a + b - c$$ is

Since $$abc = 23(a+b+c)$$ and $$23$$ is prime, $$23 \in \{a,b,c\}$$. Let $$a = 23$$:

$$bc = 23 + b + c \Rightarrow (b-1)(c-1) = 24$$.

Prime pairs: $$(b,c) = (3, 13)$$ or $$(5, 7)$$.

For maximum $$a + b - c$$: put the smallest prime as $$c$$.

Max $$= 33$$.

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