# GCD and Primes | PRMO 2017 | Question 29

Try this beautiful problem from the PRMO, 2017 based on GCD and Primes.

## GCD and primes - PRMO 2017

For each positive integer n, consider the highest common factor $h_n$ of the two numbers n!+1 and (n+1)! for n<100, find the largest value of $h_n$.

• is 107
• is 97
• is 840
• cannot be determined from the given information

### Key Concepts

GCD

Primes

Inequalities

PRMO, 2017, Question 29

Elementary Number Theory by David Burton

## Try with Hints

First hint

n! +1 is not divisible by 1,2,.....,n (n+1)! divisible by 1,2,....,n then $hcf \geq (n+1)$ and (n+1)! not divisible by n+2, n+3,...... then hcf= (n+1)

Second Hint

let n=99, 99! +1 and (100)! hcf=100 not possible as 100 |99! and 100 is non prime

Final Step

let n=97 96! + 1 and 97! both divisible by 97 then hcf=97.

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