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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 TI-89, 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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# greatest counting number divides 364, 414, 539 with same remainder

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