# Finding a perfect square

• Jun 19th 2011, 11:37 AM
skyd171
Finding a perfect square
What is the smallest positive integer k such that the product 1575 * k is a perfect square?

How does one solve this problem ?
• Jun 19th 2011, 11:44 AM
Moo
Re: Finding a perfect square
Hello,

First find the prime decomposition of 1575, then multiply by some prime numbers in order to make the final product a product of even powers of prime numbers...

Example :
smallest k such that 450*k is a perfect square.
$450=2*3^3*5^2$

In order to have an even (hence a square) power of 2, we have to multiply by 2.
In order to have an even power of 3, we have to multiply by 3.
There's no need to do it for 5, because it's already an even power.

So k=2*3=6 is the number we're looking for.

Now try to do it for 1575.
• Jun 19th 2011, 11:45 AM
SpringFan25
Re: Finding a perfect square
the prime factorisation of 1575 is $1575 = 3^2 \times 5^2 \times 7$

so

$1575 \times k = 3^2 \times 5^2 \times 7 \times k$

square numbers have only even powers in their prime factorisation. We need to find the smallest factor k which makes all the powers even. This is (by inspection) 7.

$1575 \times 7 = 3^2 \times 5^2 \times 7 \times 7 = 3^2 \times 5^2 \times 7^2 = 105^2$

Edit: Didn't see previous post.
• Jun 19th 2011, 01:09 PM
Also sprach Zarathustra
Re: Finding a perfect square
Quote:

Originally Posted by skyd171
What is the smallest positive integer k such that the product 1575 * k is a perfect square?

How does one solve this problem ?

It follows from Fundamental theorem of arithmetic - Wikipedia, the free encyclopedia.