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

## Check the Answer

But try the problem first…

Answer: is 97.

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.

## Other useful links

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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