Find the smallest positive integer a for which 2a is a perfe

Find the smallest positive integer a, for which 2a is a perfect square and 3a is a perfect cube.

Solution

for a positve integer a,there should be a perfect square 2a and perfect cube 3a.

Let suppose x,y is a arbitary numbers then,

2a= (2x)²=4x²

a=2x²

and

3a = (3y)²=27y³

a=9y³

so

2x²=9y³

since L.H.S is always even then 9y³ should be also even so value of y cant be odd.

smallest even value of y is 2.

so

y=2

    2x²=9y³

  2x²=9*2³=9*8=72

x=6

so

a= 2x²=2*36=72

a=72 answer