# What is the greatest square number that is a factor of 72 and a whole number

Another method is to look at the prime factorization of 72, which is `72=2^3*3^2.`

Now pull out the largest square that can be formed from all the factors. This is `2^2*3^2=4*9=36.`

The quick way to see this is to take each factor to its highest even power that occurs (something to an even power is always a square number) and then multiply them. To see this with a new example, let's find the largest square that is a factor of 1728. First do the prime factorization: `1728=2^6*3^3. `

The highest even power of the factor 2 is 6, and the highest even power of the factor 3 is 2. Thus the largest square dividing 1728 is `2^6*3^2=576. `

The advantage of this method is that once you have the prime factorization, the answer can be read off immediately. The real work goes into finding this factorization in the first place!

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First, we can start considering all pairs of numbers that multiply to 72:

1*72

2*36

3*24

4*18

6*12

8*9

We don't need to consider the negative numbers since we are talking about the greatest square number. Squared numbers are always positive.When we considers these pairs of numbers, we normally don't consider making one of the numbers in these pairs a fraction or decimal. If we do consider one of the numbers in these pairs a fraction or decimal, then the greatest square number would reach to infinity.

So, then, going by these numbers, we compare them to the list of squared numbers:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, etc.

(from 1 squared, 2 squared, 3 squared, 4 squared, etc.)

The greatest number that occurs in both is 36.

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