38
Why?
Consider the three odd composites 9,25,35. These are, respectively, 0,1,2(mod3).
Thus, if n is an even number, one of n−9,n−25,n−35 is an odd composite divisible by 3 (well, supposing it is >3 at least). So 35+3=38 is the largest even number that qualifies!

Some special features: 38 is the last Roman numeral when written lexicographically (XXXVIII). 38 is the largest known even number which can be represented as sum of two distinct primes in only one way: 38 = 31 + 7 38 is the sum of the squares of the first three primes:
38 = 2^2 + 3^2 + 5^2 38 is magic hexagon's smallest integer constant

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