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.

Source
Suggested Reading

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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