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September 11th, 2010, 10:12 PM  #1 
Newbie Joined: Sep 2010 Posts: 6 Thanks: 0  largest integer
hi all, i want to Find the largest integer that divides 364, 414, and 539 with the same remainder in each case... can u just help me solving this problem...please provide complete explanation... 
September 11th, 2010, 10:45 PM  #2 
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0 
We have 364 = BQ_1 + R, 414 = BQ_2 + R and 539 = BQ_3 + R, where B is the largest integer we wish to determine and R is the same remainder in each case.

September 11th, 2010, 11:01 PM  #3 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 465 Math Focus: Calculus/ODEs  Re: largest integer
I wrote a short program on my TI89, and found that 25 is the largest integer that divides the three numbers with a remainder of 14 in each case. 
September 11th, 2010, 11:26 PM  #4 
Senior Member Joined: Apr 2007 Posts: 2,140 Thanks: 0 
Okay. Can you prove that B = 25 using paper and pencil?

September 11th, 2010, 11:33 PM  #5 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 465 Math Focus: Calculus/ODEs  Re: largest integer
No, and my approach was very similar to yours, but I found too many variables and not enough equations. That's why I resorted to brute force rather than finesse. I would be interested to learn how it is done with pen and paper. 
September 12th, 2010, 12:02 AM  #6 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,765 Thanks: 1014 Math Focus: Elementary mathematics and beyond  Re: largest integer
539  14 = 525, 414  14 = 400, 364  14 = 350, so 25 divides 539, 414 and 364 with a remainder of 14. Now you can check if 539  1, 414  1, 364  1; 539  2, 414  2, 364  2 and so on up to 539  13, 414  13, 364  13 have a common divisor that is greater than 25. (I used a calculator to check them). 
September 12th, 2010, 12:47 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 18,707 Thanks: 1530 
The required integer is the greatest common divisor of 414  364 = 50 and 539  414 = 125. It's easy to see why, and to find that gcd(50, 125) = 25. 

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