 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  An Atypical Evaluation (Posted on 2006-08-13) Determine all possible positive integers x and y such that 3x-2y=17.

 Submitted by K Sengupta Rating: 4.0000 (2 votes) Solution: (Hide) (x,y) = (4, 6) constitutes the only possible solution to the equation under reference. EXPLANATION: Let, if possible, at least one of x and y be odd. If x is odd, then 2^y = (-1)^y(Mod 3)= -1(Mod3) or, 2^y - 3^x = -1(Mod 3). However, 2^y - 3^x = - 17= 1(Mod 3), which is a contradiction. Accordingly, x must be even. If y is odd, then 3^x= 3 (Mod 8). Consequently, for y >=3, we observe that: 2^y - 3^x = -3(Mod 8)= 5(Mod8). By the problem, 2^y - 3^x = - 17 or, 2^y - 3^x = 7 (Mod 8). This is a contradiction.Hence, if y is odd, it follows that y must be less than 3; or, in other words, y is equal to 1. But, for y=1, we obtain 3^x = 19, which is not feasible. Consequently, both x and y must be even, so that: x=2m and y=2n, for two positive integers m and n. Accordingly, we obtain: 3^(2m) - 2^(2n) = 17 or, (3^m + 2^n)(3^m - 2^n)= 17*1 or, (3^m + 2^n, 3^m - 2^n) =(17,1); (-17, -1); (1, 17);(-1, -17) or, (3^m,2^n)=(9, 8)(-9, -8); (9, -8); (-9, 8); Or, (m,n)=(2,3) corresponding to (3^m,2^n)=(9,8) and each of the other pairs for (3^m, 2^n) contains at least one negative integer, which is a contradiction, so that: (x,y) = (4,6) which constitutes the only possible solution to the equation under reference. ----------------------------------------------------------------------------------- For a different methodology leading to the same result, refer to the solution submitted by Dennis in this location. Comments: ( You must be logged in to post comments.)
 Subject Author Date a solution Dennis 2006-09-14 10:44:09 Addition to Previous Lillian 2006-08-28 14:30:35 Solution Lillian 2006-08-28 14:20:59 Analytical solution Hugo 2006-08-14 11:01:45 No Subject Gregor 2006-08-14 08:22:20 computer exploration--possible spoiler Charlie 2006-08-13 12:00:31 Please log in:
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