gcd

The gcd is 34.

What is the greatest common divisor of 121^2 + 233^2 + 345^2 and 120^2 + 237^2 + 348^2?  

Just checking if the first "Guest" answer is even plausible. 

    121²    +    233²    +    345²  

= 14,641  +  54,289  +  119,025  

=  187,955    This is not evenly divisible by 34 so the first "Guest" is a liar. 

GCD {[121^2 + 233^2 + 345^2] ,  [120^2 + 237^2 + 348^2]}

Prime factorizations

187955 = 5×37591 (2 distinct prime factors)
191673 = 3^3×31×229 (5 prime factors, 3 distinct)

Therefore: GCD = 1