So you have to find the lagest prime number under 28, beacuse if you multiple 2 numbers, even if they are prime, you will get a compostie number. And we know that the largest prime number less than 28 is 23, so from here the answere is 23.
Since both have 27 factorial, we can factorize to get 27! (1+28) = 27! * 29.
The largest prime that divides 27! + 28! is 29.
28! means 28 factorial which is 28*27*26...*1.
27! is 27*26*25...*1.
So we can take out 27!.