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September 21st, 2013, 02:19 PM  #1 
Newbie Joined: Sep 2013 Posts: 5 Thanks: 0  A really hard math problem "Prove that if..."
Prove that if a b c are integers and satisfy the equation (a + 3)^2 + (b + 4)^2 ?(c + 5)^2 = a^2 + b^2 ? c^2 the common value of both sides is a perfect square. I would be very grateful if someone can help me 
September 21st, 2013, 04:01 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,821 Thanks: 1047 Math Focus: Elementary mathematics and beyond  Re: A really hard math problem "Prove that if..."
Expanding the left hand side and canceling like terms gives 6a + 8b  10c = 0, so a and b are multiples of 5. Since a and b are multiples of 5, (4a  3b)/5 is always an integer, so a² + b²  c² is a perfect square. 
September 21st, 2013, 04:26 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,162 Thanks: 1638 
It is not necessary that a and b are multiples of 5, but it's easy to see that 4a  3b is a multiple of 5.

September 21st, 2013, 05:17 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,821 Thanks: 1047 Math Focus: Elementary mathematics and beyond  Re: A really hard math problem "Prove that if..."
Thanks for the correction. Can you explain why 4a  3b is a multiple of 5?

September 21st, 2013, 05:44 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 19,162 Thanks: 1638 
You showed that ((4a  3b)/5)² = a² + b²  c² (an integer). Alternative ending: a² + b²  c² ? (2a + b  2c)²  (a  c)(3a + 4b  5c) = (2a + b  2c)². 
September 21st, 2013, 05:48 PM  #6 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,821 Thanks: 1047 Math Focus: Elementary mathematics and beyond  Re: A really hard math problem "Prove that if..."
Thanks. 
September 22nd, 2013, 03:21 AM  #7 
Newbie Joined: Sep 2013 Posts: 5 Thanks: 0  Re: A really hard math problem "Prove that if..."
Thanks Guys @skipjack I do not fully understand your alternative ending; can you explain me that? 
September 22nd, 2013, 09:04 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 19,162 Thanks: 1638 
a² + b²  c² ? (2a + b  2c)²  (a  c)(3a + 4b  5c) holds for any values of a, b and c. The righthand side reduces to its first term, which is a perfect square, because the equation given in the problem can be simplified to 3a + 4b  5c = 0, which implies that the second term is zero.

September 22nd, 2013, 09:35 AM  #9 
Newbie Joined: Sep 2013 Posts: 5 Thanks: 0  Re: A really hard math problem "Prove that if..." 

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