What is the minimum value of
$ef$
?

(Here is an example set of numbers that satisfies the equation:
$a=2,b=53,c=5,d=23,e=13,f=17$
)

The answer is 55.

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quick check using Mathematica

p = Prime; z = {};

For[a = 1, a < 2, a++,

For[c = 1, c < 3, c++,

For[e = 1, e < 10, e++,

For[f = 1, f < 10, f++,

For[b = 1, b < 20, b++,

For[d = 1, d < 20, d++,

If[p@a

p@b + p@cp@d == p@e*p@f && a != b != c != d != e != f, Print[p@a, " ", p@b, " ", p@c, " ", p@d, " ", p@e, " ", p@f]; Abort[]]]]]]]]returns 2 17 3 7 5 11