First suppose aq - bp = 1, then aqc-bpc = c; combine this with ax + by = c; therefore a (qc − x) − b ( pc + y) = 0; therefore qc − x = bt, pc +y=at, where t is some integer. Hence Solutions will be found by giving to t, if possible, positive pc integral values greater than and less than Next suppose aq — bp = - a qc ·1, then aqc – bpc = c; combine this with ax + by = c, therefore a (x + qc) − b (pc − y) = 0. Hence y=pc-at. Solutions will be found by giving to t, if possible, positive до integral values greater than and less than b pc α 630. To find the number of solutions in positive integers of the equation ax + by = c. α Let be converted into a continued fraction, and let α be the convergent immediately preceding; then aq − bp = ± 1. Suppose aq-bp = 1. Then by the preceding Article, Then the least admissible value of t is m + 1, and the greatest is n; thus the number of solutions is n that is, - m, qc pc +f-g, that is,+f-g. And as this result must be an integer it must If we Then ƒ= 0; thus when tm the value of y is zero. include this solution the number of solutions is equal to the с greatest integer in + 1; if we exclude this solution the number ab of solutions is equal to the greatest integer in III. Suppose an integer. с ab If we Then g= 0; thus when t = n the value of x is zero. include this solution the number of solutions is equal to the с greatest integer in + 1; if we exclude this solution the number ab of solutions is equal to the greatest integer in. Then ƒ 0, and g = 0; ab thus when t = m the value of y is zero, and when t = n the value of x is zero. If we include these solu tions the number of solutions is equal to с + 1; if we exclude ab these solutions the number of solutions is - 1. ab с Thus the number of solutions is determined in every case. Similar results will be obtained on the supposition that aq-bp = -1. 631. To solve the equation ax + by + cz = d in positive integers we may proceed thus: write it in the form ax + by=d- cz, then ascribe to z in succession the values 1, 2, 3, .... and determine in each case the values of x and y by the preceding articles. ...... 632. Suppose we have the simultaneous equations ax + by + cz= =d, ax + b'y + c'z = d'; eliminate one of the variables, z for example, we thus obtain an equation connecting the other two variables, Ax + By = C, suppose. Now if A and B contain no common factors except such as are also contained in C, by proceeding as in the previous articles, we may obtain Substitute these values in one of the given equations, we thus obtain an equation connecting t and z, which we may write At + B'z C'. From this, if A' and B' contain no common factors except such as are also contained in C', we may obtain = Substitute the value of t in the expressions found for x Hence we obtain for each of the variables x, y, an expression of the same form as that already obtained for z. EXAMPLES OF INDETERMINATE EQUATIONS. Solve the following six equations in positive integers : Find the general integral values in each of the following four equations, and the least values of x and y which satisfy each: 11. In how many ways can £500 be paid in guineas and five-pound notes? 12. In how many ways can £100 be paid in guineas and crowns? 13. In how many ways can £100 be paid in half-guineas and sovereigns? 14. In how many ways can £22. 3s. 6d. be paid with French five-franc pieces (value 4s. each), and Turkish dollars (value 3s. 6d. each)? 15. In how many ways can 19s. 6d. be paid in florins and half-crowns? 16. If there were coins of 7 shillings and of 17 shillings, in how many ways could £30 be paid by means of them? 17. What is the simplest way for a person who has only guineas to pay 10s. 6d. to another who has only half-crowns? 18. Supposing a sovereign equal to 25 francs, how can a debt of 44 shillings be most simply paid by giving sovereigns and receiving francs? 19. Divide 200 into two parts, such that if one of them be divided by 6 and the other by 11, the respective remainders may be 5 and 4. 20. How many crowns and half-crowns, whose diameters are respectively 81 and 666 of an inch, may be placed in a row together, so as to make a yard in length? 21. Find n positive integers in arithmetical progression whose sum shall be n2; shew that there are two solutions when n is odd. 22. What is the least number which divided by 28 leaves a remainder 21, and divided by 19 leaves a remainder 17 ? 23. Find the general form of the numbers which divided by 3, 5, 7, have remainders 2, 4, 6, respectively. 24. What is the least number which being divided by 28, 19 and 15, leaves remainders 13, 2 and 7? 25. Solve in positive integers 17x+23y+ 3z = 200. 26. Find all the positive integral solutions of the simultaneous equations 27. In how many ways can a person pay a sum of £15 in half-crowns, shillings, and sixpences, so that the number of shillings and sixpences together shall equal the number of half crowns? 28. Find in how many different ways the sum of £4. 16s. can be paid in guineas, crowns, and shillings, so that the number of coins used shall be exactly 16. 29. How can £2. 4s. be paid in crowns, half-crowns, and florins, if there be as many crowns used as half-crowns and florins together? 30. What is the greatest sum of money that can be paid in 10 different ways and no more, in half-crowns and shillings? 31. The difference between a certain multiple of ten and the sum of its digits is 99; find it. 32. The same number is represented in the undenary and septenary scales by the same three digits, the order in the scales being reversed and the middle digit being zero; find the number. 33. A number consists of three digits which together make up 20; if 16 be taken from it and the remainder divided by 2 the digits will be inverted; find the number. 34. Find a number of four digits in the denary scale, such that if the first and last digits be interchanged, the result is the same number expressed in the nonary scale. Shew that there is only one solution. = 35. A farmer buys oxen, sheep, and ducks. The whole number bought is 100, and the whole sum paid £100. Supposing the oxen to cost £5, the sheep £1, and the ducks 1s. per head; find what number he bought of each. Of how many solutions does the problem admit? 36. Find three proper fractions in Arithmetical Progression whose denominators shall be 6, 9, 18, and whose sum shall be 23. 37. Three bells commenced tolling simultaneously, and tolled at intervals of 25, 29, 33 seconds respectively. In less than half |