Which integer between 2 and 200000000000000 is a perfect square a perfect cube and a perfect fifth power?

Answer 1

The integer that is a perfect square, perfect cube, and perfect fifth power is 64. This is because 64 is equal to 8^2, 4^3, and 2^6, making it a perfect square, cube, and fifth power. It is the only integer between 2 and 200000000000000 that satisfies all three conditions.

Answer 2

If a number is a perfect square, then its prime factors can be split into two identical groups: if a perfect cube, three groups; if a perfect fifth power, five groups. The best way to attain this sort of 'symmetry' is to only use one prime factor. Since we are looking for small (ish) solutions, we'll use 2. It only remains to decide how many 2's to multiply together. The number needs to be divisible by 2, 3 and 5. We are looking for the lowest common denominator of 2, 3 and 5, which is 30. 22x3x5 = 230 = 1073741824 = (22x3)5 = (22x5)3 = (23x5)2 The next smallest solution is given by 330 = 205891132094649, which is just a little bit bigger than 2 x 1014. So the question is correct in assuming that only one answer exists.