Prove that there are infinitely many numbers n such that 2n is a perfect square, 3n is a perfect cube, and 5n is perfect 5th power.

Level: Junior

Take n = (2^15)*(3^20)*(5^24)*(k^30) for positve integer k. Thus
2A = [(2^8)*(3^10)*(5^12)*(k^15)]^2 is a perfect square,
3A = [(2^5)*(3^7)*(5^8)*(k^10)]^3 is a perfect cube,
5A = [(2^3)*(3^4)*(5^5)*(k^6)]^5 is a perfect 5th power.
Hence there are infinitely many numbers n satisfying the given conditions.