It is given that am⋅bn=144145, where a>1 and b>1. 144 can be written as 144=24×32 Hence, am⋅bn=144145 can be written as am⋅bn=(24×32)145=2580×3290 We know that 3290 is a natural number, which implies it can be written as a1, where a>1 Hence, the least possible value of m is 1 . Similarly, the largest value of n is 580 . Hence, the largest value of (n−m) is (580−1)=579 The correct option is D