# How to solve it

1. Mar 27, 2005

### bomba923

"Find a positive integer 'm' such that m/2 is a perfect square and m/3 is a perfect cube."

How to solve it?

2. Mar 27, 2005

### matt grime

consider m's prime decomposition

3. Mar 27, 2005

### dextercioby

The smallest one can find is

$$M=3^{4}\cdot 2^{3}=648$$

Daniel.

4. Mar 27, 2005

### bomba923

Prime factorization::thanks

Now just one more question: Is it impossible to construct an equilateral triangle using only lattice points?---(i.e., points with integer coordinates)
My work says no---but can it be done?

5. Mar 27, 2005

### dextercioby

Nope,if one of the sides is || with one of the axis of coordinates...The height of the triangle is

$$h=\frac{l\sqrt{3}}{2}$$...(that justifies it)...

As for the general case,it is a system of 3 equations with 6 unknowns...

Daniel.