Find the smallest positive integer n such that n/2 is a perfect square, n/3 is a perfect cube and n/5 is a perfect fifth power.
Given: n/2 is a perfect square
so, let
n= 1 , then n/2 = 1/2 which is not a perfect square,
n=2, then n/2 = 2/2 = 1 which is not a perfect square,
and so on till
n= 8, then n/2 = 8/2 = 4 and root of 4 = 2 , which is an integer
so, 2 is the smallest +ve integer for n/2 to be perfect square.
Now, also, its given that, n/3 is a perfect cube
let n= 1, then n/3 = 1/3 which is not a perfect cube,
and so on till
n=27, then n/3 = 27 /3 = 9 not a perfect cube,
n= 81 ,then n/3 = 81/ 3= 27 , which is a perfect cube.
so, 27 is a smallest +ve integer for n/3 to be perfect cube.
now, for n/5 to be perfect fifth power.
let n= 1, then n/5 = 1/5 , is not a perfect fifth power
and so on till ...
n= 15625, then n/5 = 15625/ 5 = 3125 =5*5*5*5*5