  We consider a few modifications of the Big prime modular $\gcd$ algorithm for
polynomials in $\Z[x]$. Our modifications are based on bounds of degrees of
modular common divisors of polynomials, on estimates of the number of prime
divisors of a resultant and on finding preliminary bounds on degrees of common
divisors using auxiliary primes. These modifications are used to suggest
improved algorithms for $\gcd$ calculation and for coprime polynomials
detection. To illustrate the ideas we apply the constructed algorithms on
certain polynomials, in particular, on polynomials from Knuth's example of
intermediate expression swell.
