In this task, you need to answer the given multiple-choice question on geometry. Classify your answers into 'a', 'b', 'c', 'd', and 'e'.

Problem: a perfect square is defined as the square of an integer and a perfect cube is defined as the cube of an integer . how many positive integers n are there such that n is less than 100,000 and at the same time n is a perfect square and a perfect cube ?
Options: a ) 3 , b ) 4 , c ) 5 , d ) 6 , e ) 7
d

Problem: a crate measures 2 feet by 8 feet by 12 feet on the inside . a stone pillar in the shape of a right circular cylinder must fit into the crate for shipping so that it rests upright when the crate sits on at least one of its six sides . what is the radius , in feet , of the pillar with the largest volume that could still fit in the crate ?
Options: a ) 2 , b ) 4 , c ) 6 , d ) 8 , e ) 12
a

Problem: a certain rectangular window is three times as long as it is wide . if its perimeter is 32 feet , what are its dimensions in terms of length by width ?
Options: ['a ) 12 by 2', 'b ) 11 by 3', 'c ) 10.5 by 3.5', 'd ) 12 by 4', 'e ) 9 by 3']
d