7, 119, 140, 625 = 5×ll253=3×755
The reason why the smallest solution is in the billions, however, is not hard to see. Any solution n has to have the form 3r5s m for some positive powers r and s and where the remaining prime factors are collected together into a single integer m that is not divisible by 3 0r 5. If we first focus on the possible values of r, we observe that since n is 5 times a cube, the exponent r must be a multiple of 3, and since n is 3 times a 5th power, the number r-1has to be a multiple of　5. The smallest r that satisfies both these conditions simultaneously is r=6. In the same way, the exponent s has to be a multiple of 5, while s--1 has to be a mutiple of 3 and the least s that fits the bill is s=10. To make n as small as possible,we takem=1 andson=36×510=3(3×52)5=3×755,so that n is indeed 3 times a 5th power and at the same time n=5(32×53)3=5×11253, and so nis also 5 times a cube.