1. ## Smallest Integer

What is the smallest integer greater than 1 that is both the square of an integer and the cube of an integer?

2. ## Entered By Symmetry

For the record, I deleted this message. How it got to you so quickly I don't know since I deleted it almost as soon as I posted it. I realized that I had read the question wrong: there is nothing in the problem that says the number has to be the square and cube of the SAME integer as I supposed when I did the above analysis.

I believe the answer is 64 = 8^2 = 4^3, but I can only find that by trial and error. I presume someone will come along and show you how to do this.

3. Originally Posted by symmetry
What is the smallest integer greater than 1 that is both the square of
an integer and the cube of an integer?
---End Quote---
You have x^3 = x^2.

So
x^3 - x^2 = 0

x^2(x - 1) = 0

So either x^2 = 0, which implies x = 0 or x - 1 = 0 which implies x =
1.

Since x = 0 and x = 1 are the only solutions there is no number (much
less an integer!) greater than 1 with this property.

-Dan
For the record, I deleted this message. How it got to you so quickly I don't know since I deleted it almost as soon as I posted it. I realized that I had read the question wrong: there is nothing in the problem that says the number has to be the square and cube of the SAME integer as I supposed when I did the above analysis.

I believe the answer is 64 = 8^2 = 4^3, but I can only find that by trial and error. I presume someone will come along and show you how to do this.

-Dan