January 2017 Problem

x is a prime number greater than 3. Prove that 2x²+190 is a multiple of 48.

Dear  visitors and friends of the site. Thank you for your solutions. Sadly, this time no correct answers have been submitted.

Here is the solution:

Our number is 2(x² - 1+96) .  Then x² - 1 =(x-1)(x+1).

As x is prime, x-1 and x+1 are even and one of them s a multiple of 4. Hence x² - 1 is a multiple of 8.

Numbers x-1,x,x+1 have among them one multiple of 3 (this is not x). Thus, x² - 1 is a multiple of 24 and hence our original expression is a multiple of 48.

Thank you again and see you in February. Serge Hazanov