I love your 'cut the knot website', which is indeed both accurate, serious, and authoritative in your mathematical literature which I find very suitable for both the young's and old readers.

I have a question and wonder whether you have encountered any simpler proof which comes a geometrical interpretation of the below theorem found in your webpage.

gcd(N,M) * lcm(N,M) = N * M

Noting the fact that this theorem is only true for set of two numbers ( only true for two-dimensional situation)I wonder whether

we can use the area of a rectangle with dimension of their length to represent N and M.

Thanks very much

Gratefully Yours

Lawrence

Perhaps, you do not want to stack the squares but emphasize the division lines in the N×M rectangle. Once the horizontal lines and on a separate diagram the vertical ones (keeping the lines defined by the squares in the other dimension.) The big rectangle will be tesselated with smaller rectangles of which one side equals to that of gcd. The total number of the small rectangles will be the same in both cases and having the two tesselations (vertical and horizontal) will show that this total is a multiple of both N and M.

Will think of making this into an applet. Thank you for the idea.

Noting the fact that this theorem is only true for set of two
numbers (only true for two-dimensional situation) I wonder whether
we can use the area of a rectangle with dimensions of their length and breadth represented by loga to the base a and logb to the
base a. I also wonder whether it is possible to do some form of transformation to the area under the log curve bounded by x=a and x=b to a rectangle whose sides is always a reciprocal of each other so that their product equal 1; Thanks very much
Gratefully Yours
Lawrence