Jim makes the following conjecture: other than 1, there are no numbers less than 100 that are perfect squares and perfect cubes. what is a counterexample that proves his conjecture false?
The number 64 is a counterexample to Jim's conjecture because it is the square of 8 and the cube of 4.
A fast way to solve this problem is to list out the perfect cubes that are less than 100 and check if any of them are also perfect squares. A perfect square is a number which can be written as another number multiplied by itself, and a perfect cube is one which can be written as a number multiplied by itself twice.
The perfect cubes less than 100 are as follows:
1*1*1 = 1
2*2*2 = 8
3*3*3 = 27
4*4*4 = 64
Now we can calculate perfect squares until we find one in this list
1*1 = 1
2*2 = 4
3*3 = 9
4*4 = 16
5*5 = 25
6*6 = 36
7*7 = 49
8*8 = 64
We see that 64 is in the list of perfect squares and perfect cubes, so this is a counterexample to Jim's conjecture.
