If \( a=\prod_i p_i^{a_i},\ b=\prod_i p_i^{b_i}\) then \( a/(a,b)=\prod_i p_i^{c_i}\)
where \( c_i=\max\{ a_i-b_i,0\} \); we write \( \Delta(a,b)\) for the vector \( c\). We study here the size of the set
\( \Delta(A,B)\).
Assuming the \( abc\)-conjecture we show that there are only finitely many powerful binomial coefficients \( \binom nk\) with \( 3\leq k\leq n/2\), since if \( q^2\) divides \( \binom nk\) then \( q\ll n^2 \binom nk^{o(1)}\). Unconditionally we show that there are \( N^{1/2+o(1)} \) powerful binomial coefficients in the top \( N\) rows of Pascal's triangle.