Find an integer n such that n/2 is a square, n/3 is a cube, and n/5 is a fifth power.
Is there a way to do this without using a computer? If so, can anyone give me a hint. It's killing me.
Hello, paupsers!
Since is divided by 2, 3, and 5,
. . we assume that has factors of 2, 3, and 5 (at least).
Let . for some positive integers
is a square.
We have: .
. . where: . are all even. .[1]
is a cube.
We have: .
. . where: . are all mutiples of 3. .[2]
is a fifth power.
We have: .
. . where: . are all multiples of 5. .[3]
Combining [1], [2] and [3], we have:
. This is satisfied by: .
.This is satisfied by: .
. This is satisfied by: .
Therefore, the least positive integer is:
. . . .