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1 solution
To maximize the lcm of the integer variables a, b, c, d with a given sum p,
the values chosen for a, b, c, d should be coprime
the values chosen for a, b, c, d should be close to p/4.
Here, p = a + b + c + d = 1002, so the values chosen for a, b, c, d must all be odd values, close to 250.5.
In the range 244 - 257, the coprimes are
247 (13x19), 251 (prime), 253 (11x23) and 257 (prime)
either 245 (5x7^2) or 255 (3x5x17)
either 249 (3x83) or 255 (3x5x17).
There are two combinations of these values which add up to 1002. These combinations are, together with their lcm's:
245, 247, 253, 257 (3934745815)
245, 249, 251, 257 (3935249535)
So the solution is 3935249535 (not surprisingly, as 249 and 251 are closer to 250.5 than 247 and 253).