A) Find the integers x, y, and z that satisfy the equation 6x+15y+20z=1
B) under what conditions on a,b, and c is it true that the equation ax+by+cz=1 has a solution? Descride a general method of finding a solution when one exists.
C) use this method to find integers that satisfy 155x+341y+385z=1
One approach: Joe came back from a stamp collectors gathering and told his sister, Jill, that he bought a hundred stamps. Joe said he bought the stamps at four different prices: $0.59, $1.99, $2.87, and $3.44 each. Jill asked, "How much did you pay altogether?" Joe replied, "One hundred dollars exactly." How many stamps did Joe buy at each price?
Given: (1)--W + X + Y + Z = 100 and (2)--.59W + 1.99X + 2.87Y + 3.44Z = 100.
1--Multiply (2) by 100--->59W + 199X + 287Y + 344Z = 10000. (3)
2--Multiply (1) by 59----->59W + 59X + 59Y + 59Z = 5900. (4)
3--Subtract (4) from (3)--->140X + 228Y + 285Z = 4100. (5)
4--Divide (5) through by 140--->X + Y + 88Y/140 + 2Z + 5Z/140 = 29 + 40/140.
5--Solving for X = 29 - Y - 2Z - (88Y + 5Z - 40)/140. (6)
6--Set (88Y + 5Z - 40)/140 = u = an integer.
7--Rearranging, 140u = 88Y + 5Z - 40. (7)
8--Dividing (7) through by 5--->28u = 17Y + 3Y/5 + Z - 8. (8)
9--Solving for Z, Z = 28u - 17Y - 3Y/5 + 8. (9)
10--With 3Y/5 = integer, multiply by 2 yielding 6Y/5. (10)
11--Dividing (10) out yields 6Y/5 = Y + Y/5 where Y/5 must be an integer also.
12--Set Y/5 = v whence Y = 5v. (11)
13--Substituting (11) into (8)--->Z = 28u - 85v - 3v + 8 = 28u - 88v + 8. (12)
14--Substituting (11) and (12) into (6)--->X = 29 - 5v -56u + 176v -16 - (440v + 140u - 440v + 40 - 40)/140. (13)
15--Simplifying (13)--->X = 29 - 5v -56u + 176v - 16 - u = 13 + 171v - 57u. (13)
16--Substituting (11), (12), and (13) into (1)---> W + 13 + 171v - 57u + 5v + 28u - 88v + 8 = 100. (14)
17--Simplifying (14)---> W = 29u - 88v + 79. (14)
18--From (11) v =/> 1 and from (12) u =/> 3.
19--Trying v = 1 and u = 3, W = 78, X = 13, Y = 5, and Z = 4.
20--Checking--->78 + 13 + 5 + 4 = 100. Okay.
21--Checking--->78(.59) + 13(1.99) + 5(2.87) + 4(3.44) = 46.02 + 25.87 + 14.35 + 13.76 = 100. Okay.
22--Trying v = 2 and u = 3, X = 184 exceeding total of 100, therefore invalid.
23--Trying v = 1 and u = 4, X = negative number, therefore invalid.
24--All other values of u and v produce invalid results.
25--Therefore W = 78, X = 13, Y = 5, and Z = 4 is the only solution.
I hope this has been of some interest to you. The solution of problems of this type, N unknowns with N-1 equations, is not difficult and can usually be solved by this successive reduction method. If you are at all interested in exploring the methods for solving these types of equations with two or more unknowns, I offer you the following references:
1--Number Theory and its History--by Oystein Ore--Dover Publications, Inc., 1976
2--Mathematical Brain Teasers--by J.A.H. Hunter--Dover Publications, Inc., 1976
3--Mathematics for the Nonmathemetician by Morris Kline, Dover Publications, Inc., 1985.
4--Recreations in the Theory of Numbers by Albert H. Beiler, Dover Publications, Inc., 1964.
5--An Adventurer's Guide to Number Theory by Richard Friedberg, Dover Publications, Inc., 1968.
6--Excursions in Number Theory by C. Stanley Ogilvy and John T. Anderson, Dover Publications, Inc., 1988.
7--Elementary Theory of Numbers by William J. LeVeque,, Dover Publications, Inc., 1990.
8--Fundemental Concepts of Algebra by Bruce E. Meserve, Dover Publications, Inc., 1982.
If you are truly interested in the historical and/or further aspects of recreational mathematics, I strongly reccommend you send for the Dover Publications catalog of Math and Science books. They publish one of the widest varieties of math books I have run across, many of them, reprints of classics in the field. Their address is 31 East 2nd Street, Mineola, NY 11501, Tel. No. 516-294-7000. The linear indeterminate problem methodology is best described in the book titled, Number Theory and Its History by Oystein Ore.
There might very well be many other books available in your school, or local, library or through your teachers. These, will probably be able to give you additional information on the specific methods for solving Linear Diophantine problems. If you are interested in exploring other problems of this type, with 2, 3, and 4 unknowns, I offer you some problems below to pursue on your own. Some of them were used in the examples above. I would first advise you to get some references from Dover or your library and learn the procedure unless you have been able to learn it from the above examples. The methods allow you to solve many problems that, at first glance, appear unsolvable.
