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.

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