105=3×5×7 Number of factors =2×2×2=8 Hence, required number of pairs =8∕2=4 Detailed Explanation: m2+105=n2 ⇒n2−m2=105 ⇒(n−m)(n+m)=105 Since m and n are positive integers, (n−m)<(n+m) Splitting 105 in two factors, we get For (n−m)=1 and (n+m)=105,(m,n)=(52,53) ⇒(n−m)(n+m)=3×35 For (n−m)=3 and (n+m)=35,(m,n)=(16,19) ⇒(n−m)(n+m)=5×21 For (n−m)=5 and (n+m)=21,(m,n)=(8,13) ⇒(n−m)(n+m)=7×21 For (n−m)=7 and (n+m)=21,(m,n)=(4,11) Hence there are four pairs.