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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!
First, we can start considering all pairs of numbers that multiply to 72:
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.
First list all of the factors of 72
72 * 1
36 * 2
24 * 3
18 * 4
12 * 6
8 * 9
The factors with the perfect squares are 4, 9, and 36
So the greatest square number that is a factor of 72 is 36
Perfect squares are whole numbers that are squares of whole numbers. So basically, you are looking for a perfect square that is a factor of 72. First we should list the factors of 72. We have 72 and 1, 24 and 3, 18 and 4, 12 and 6, 8 and 9. Among these factors the perfect squares are 4, 9, and 36. So the greatest square of a whole number that is a factor of 72 is 36.
First, I would divide 72 by whole numbers, starting with 1 and going up, until the numbers begin to repeat (they repeat starting at 9).
Now you have all you factors that can go into 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. The highest factor of 72 that is also a perfect square (a number that when put under a square root is a whole number) is 36, so that would be your answer.
When I see a problem like this, I first divide 72 by single digit numbers.
72/2 = 36
72/3 = 24
72/4 = 18
72/5 = Not a whole number
72/6 = 12
72/7 = Not a whole number
72/8 = 9
Since 9 is the next single digit, I know I am done dividing.
By doing this, you get all the factors of 72 (excluding 1 and 72). Looking at these factors, your next step is to find the integer that can be square-rooted! Be sure to start from the largest number since you are looking for the greatest square value.
Your answer should be 36.
Alternatively, you could find the greatest perfect square out of the factors of 72.
A perfect square made by squaring a whole number, and you can see that the only perfect squares in the factors of 72 are: 4, 9, and 36. So your answer can only be one of these numbers. Out of the three, 36 is the greatest, and that would be the answer you are looking for.
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