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Prime Factors and Square Products

Date: 10/05/2003 at 15:59:25
From: Alexander
Subject: Creating a square number

What is the smallest number that you can multiply by 540 to make a
square number?

I know the answer is 15, which I found by trial and error.  That is, I
worked out the prime factors and they are 2^2, 3^2, and 5.  Juggling
these brings me to 90 x 90 = 8100, which is 540 x 15.

What I cannot work out for numbers other than 540 is the correct
method of juggling.  My working was random and I'd love to be able to
apply this to other numbers. 


Date: 10/05/2003 at 19:20:29
From: Doctor Luis
Subject: Re: Creating a square number

Hi Alexander,

You were on the right track.  The prime factorization of 540 is

      540 = (2*2)*(3*3*3)*(5) 

          = 2^2 * 3^3 * 5^1

Now, when you square a number, the exponents get multiplied by two.
Therefore, the prime factors of a square number appear an even number
of times.

So, to make the product of 540 and something else a square number, you
have to make the prime factors of the product appear an even number of

You can see that there are already two 2's, so you don't worry about
the 2.  But there are three 3's and only one 5.  That's why you need
an extra 3 (so there are four instead of three) and an extra 5 (so you
have two instead of just one).  Of course, 3*5 = 15, which was your guess.

  540 * (3*5) = 2^2 * 3^4 * 5^2

Since the exponents of the prime factors are all even, we can also
find what number it squares

  540 * 15 = 2^2 * 3^4 * 5^2

           = (2 * 3^2 * 5)^2

           = (10 * 9)^2

           = 90^2

A square number, just like you wanted!

Hopefully, you can now apply this method to numbers other than 540.

Let us know if you have any more questions.

- Doctor Luis, The Math Forum 

Date: 10/07/2003 at 16:34:42
From: Alexander
Subject: Thank you (Creating a square number)

Thank you, Doctor Luis, for your help.  I had fun working through
other numbers and trying out cubes. 

Associated Topics:
High School Puzzles
High School Square & Cube Roots

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