What is the smallest positive integer n such that 2n is a perfect square 3n is a perfect cube and 4n is a perfect fourth?

Answer 1

To find the smallest positive integer ( n ) such that ( 2n ) is a perfect square, ( 3n ) is a perfect cube, and ( 4n ) is a perfect fourth, we analyze the conditions for each case using prime factorization. Let ( n = 2^a \cdot 3^b \cdot k ), where ( k ) is coprime to 2 and 3.

1. For ( 2n ) to be a perfect square, ( a+1 ) must be even and ( b ) must be even.
2. For ( 3n ) to be a perfect cube, ( a ) must be divisible by 3 and ( b+1 ) must be divisible by 3.
3. For ( 4n ) to be a perfect fourth, ( a+2 ) must be divisible by 4 and ( b ) must be divisible by 4.

By solving these conditions simultaneously, the smallest ( n ) that meets all conditions is ( n = 108 ).