# Determine the largest prime divisor of 87! + 88!.

• January 26th 2007, 11:37 AM
ceasar_19134
Determine the largest prime divisor of 87! + 88!.
Determine the largest prime divisor of 87! + 88!.

These problems are so hard... (For me anyway.)
• January 26th 2007, 01:28 PM
galactus
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.
• January 26th 2007, 01:33 PM
CaptainBlack
Quote:

Originally Posted by galactus
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.

That is the smallest prime divisor not the largest.

RonL
• January 26th 2007, 01:40 PM
galactus
how about 47?. It's the smallest, isn't it?. It's prime and it divides 87!+88!.

But you're certainly right, Cap'N. I had a major brain fart. I was thinking of prime factors. DUH
• January 26th 2007, 02:02 PM
CaptainBlack
Quote:

Originally Posted by galactus
how about 47?. It's the smallest, isn't it?. It's prime and it divides 87!+88!.

But you're certainly right, Cap'N. I had a major brain fart. I was thinking of prime factors. DUH

Opps, sorry that's two of us:o

RonL
• January 26th 2007, 02:17 PM
Soltras
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)! )
• January 26th 2007, 02:31 PM
galactus
So I was correct. I thought this was something I had seen before, but then second-guessed myself.
• January 26th 2007, 07:23 PM
Soroban
Hello, ceasar!

Quote:

Determine the largest prime divisor of $87! + 88!$

We have: . $87! + 88!\:=\:87! + 88\cdot87!$

Factor: . $87!(1 + 88) \:=\:87!(89)$

Therefore, $89$ is the largest prime divisor.