Determine the largest prime divisor of 87! + 88!.
These problems are so hard... (For me anyway.)
If you have n!+(n+1)! and (n+2) is prime, then (n+2) is the largest prime divisor.
The next number, 89, is prime, the largest prime divisor of 87!+88! is 89.
For instance, try 9!+10!, 11 is the largest prime divisor.
29!+30!...the largest prime divisor is 31.
This isn't anything I ever tried to prove. The Hackster probably knows it or will figure it out. There's probably already a known theorem relating it.
87! + 88! = 87! ( 1 + 88 ) = 87!*(89).
89 is a prime factor.
The question is, does a higher prime divide 87!
Suppose p is prime and greater than 89.
Well 87! = 87*86*85*...*2*1
Since p doesn't divide 89, then if p divides 87!*(89), then p divides at least one of these terms in 87!.
This is impossible since p is larger than all terms appearing in the factorial.
So 89 is the largest prime factor of 87! + 88!
(this can be worked into a theorem, say if p is prime, then it's the largest prime factor of (p-1)! + (p-2)! )