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 Diophantine Linear and Quadratic (Posted on 2015-11-01)
Find all possible positive integers A, B and C - with A ≤ B, that simultaneously satisfy:
• A+B-C=12, and:
• A2 + B2 - C2 = 12

 No Solution Yet Submitted by K Sengupta Rating: 3.0000 (1 votes)

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 Corrected Analytical Solution Comment 4 of 4 |
`(1) C = A+B-12(2) Substituting in the 2nd equation and simplifying gives    -156 = 24A + 24B - 2AB(3) Solving for B gives B = (-78-12A)/(12-A)(4) Let D = 12-A.  Then B = 12 - 66/D(5) The only possible values for D are +/- 1,2,3,6,11,22,33,66.So there are 16 possible solutions.`
`They are:D	B	A	C1	-54	11	-552	-21	10	-233	-10	9	-136	1	6	-511	6	1	-522	9	-10	-1333	10	-21	-2366	11	-54	-55-1	78	13	79-2	45	14	47-3	34	15	37-6	23	18	29-11	18	23	29-22	15	34	37-33	14	45	47-66	13	78	79The only solutions with A, B and C positive and A <= B are(13,78,79), (14,45,47), (15,34,37) and (18,23,29)`

 Posted by Steve Herman on 2015-11-01 16:47:48

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