A man has a garden measuring 84 meters by 56 meters. He divides it into the minimum number of square plots. What is the length of the square plots?

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We wish to divide this man's garden into the minimum number of square plots possible. A square has all four sides with the same length.Our garden is a rectangle, so the answer is clearly not 1 square plot. If we choose the wrong length for our squares, we may end up with...

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We wish to divide this man's garden into the minimum number of square plots possible.

A square has all four sides with the same length.Our garden is a rectangle, so the answer is clearly not 1 square plot. If we choose the wrong length for our squares, we may end up with missing holes or we may not be able to fit our squares inside the garden.

So we have 84 meters in one direction and 56 meters in the other direction. When we start dividing the garden in square plots, we are "filling" those lengths in their respective directions. At each direction, there must be an integer number of squares (otherwise, we get holes or we leave the garden), so that all the square plots fill up the garden nicely. Thus, our job here is to find the greatest common divisor of 84 and 56.

For this, we prime factor both of them:


`56 = 2*2*2*7`

`84 = 2*2*3*7`

 

We can see that the prime factors and multiplicities in common are `2*2*7 = 28` . This is the desired length of the square plots.

If you wish, you can also check for how many square plots the man will have. Using the length that we found, we get `56/28=2`  by `84/28=3` square plots. Thus, the man has 6 square plots for his garden.

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