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

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