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Algebra Arithmetic Math Olympiad PRMO

GCD and Primes | PRMO 2017 | Question 29

Try this beautiful problem from the Pre-RMO, 2017 based on GCD and Primes. You may use sequential hints to solve the problem.

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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