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

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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