SOLUTION: Find both square roots of the largest perfect square that divides 8124287104.

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Question 1063715: Find both square roots of the largest perfect square that divides 8124287104.
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(52799)   (Show Source): You can put this solution on YOUR website!
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Find both square roots of the largest perfect square that divides 8124287104.
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Read your post.


Comment from student: I'm sorry, but I don't understand what you mean by read your post. Maybe explain what you mean.


My response: Do you see the word "both" in your post?

What does it mean in this context ??



Comment from student: The teacher told us that there are 2 square roots to the largest perfect square and that we are to find both of them.



My response: Great !!

I saw the solution by Edwin. Great !!


For me, the formulation would be much more natural without this word "both".



Answer by Edwin McCravy(20059)   (Show Source): You can put this solution on YOUR website!
I kept dividing by 2 until I got an odd number:

8124287104÷2 = 4062143662 
4062143552÷2 = 2031071776
2031071776÷2 = 1015535888
1015535888÷2 =  507767944
 507767944÷2 =  253883972
 253883972÷2 =  126941986
 126941986÷2 =   63470993  <--an odd number

So 8124287104 = 27∙63470993

We know that 26 = (23)2 = 82 = 64

is a perfect square that divides 8124287104.

But we don't know if there is a larger one.  So I went to

http://compoasso.free.fr/primelistweb/page/prime/liste_online_en.php

and typed in 63470993 to see if it is prime and found that it was not.

So I tried dividing it by primes 3,5,7,11,13,17,19,23,29,31,37,41

and found that 63470993 = 41∙1548073

I went to that site again and found that 1548073 is indeed a prime.

So 64 is the largest perfect square that divides 8124287104.

64 has two square roots, which are 8 and -8.

Edwin
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