1 22 23 114 &c. (2) Ans. 1 1 2 7 16 3 7 25 57 84 227 in a continued fraction, and find the convergents. Ans. The quotients are 2, 1, 2, 2, 1, The convergents are 1 3 7 10 37 84 3. Find the convergents to the continued fraction whose quotients 1 5 46 97 143 240 1103 Ans. 2 1 4 37 1 , 19' 27 100 227 4. Find a series of fractions converging to √2; also to √45. Ans. The quotients are 2, 1, 4, 1, 1, 1, 4, &c. From the last example deduce an explanation of the Julian and differs from 3.14159 by a quantity less than 0.00001. 9. The lunar month, calculated on an average of 100 years, is 10. The sidereal revolution of Mercury is 87.969255 days; and 1947 3359 INDETERMINATE EQUATIONS AND PROBLEMS. cxvii 11. Find the least fraction with only two figures in each term, ap proximating to 12. Given 26. Required x in the form of a continued fraction; and find the convergents. 16. Approximate by continued fractions to the roots of the equations, 1 it differs from the latter fraction by a quantity less than 2 × 305 × 1292 INDETERMINATE EQUATIONS AND PROBLEMS. 1. 14x-5y=7; find the least positive integral values of x and y. 2. 27x+16y= 1600; 3. 19x-117y = 11; ....... Ans. x = Ans. x 48, y = 19. Ans x 56, y=9. = 4. 3x+5y=26; find all the values of x and y in positive integers. 7. 11x+7y=108; find all the values of x and y in positive integers. Ans. x 6, y = 6. = 8. Shew that there is no solution in whole numbers for the equation 49x-35y=11. 9. Given x=4, y=9, one solution of 2x + 3y = 35. Find all the solutions in positive integers. Ans. x = 1, 4, 7, 10, 13, 16. y=11, 9, 7, 5, 3, 1. 12. 3y+4z Find all the positive integral solutions of 20x-21y=38, and 13. Find all the positive integral solutions of xy + x2 = 2x + 3y + 29. Ans. x 4, 5, y=21, 7. 14. Find all the positive integral solutions of 7xy-5x=3y+39. Ans. x=4, y=12, z=7. 15. integers. Find the number of solutions of 11x+15y=1031 in positive Ans. 7. 16. Find the number of solutions of 3x + 7y+17% = 100 in positive integers. Ans. 10. 17. Find the number of solutions of 20x + 15y + 6z=171 in positive integers. Ans. 12. their sum Find two fractions having 7 and 9 for their denominators, and 19. Find three fractions with denominators 3, 4, and 5, of which the 22. Find the least number which, upon being divided by 11, 19, and 29, gives the remainders 3, 5, and 10, respectively. Ans. 4128. 23. Find a number less than 400 which is a multiple of 7, and upon being divided by 2, 3, 4, 5, 6, always leaves 1 for a remainder. Ans. 301. 24. Shew that the solution of ax + by = c in positive integers is always possible, if c> ab− (a+b). 25. In how many different ways is it possible to pay £20 in halfguineas and half-crowns? 26. A certain sum consists of x£. y shillings, and its y £. x shillings; find the sum. Ans. 7. half of Ans. £13. 6s. 27. Find two numbers such that their sum shall be equal to the sum of their squares. 15 3 Ans. , 28. What value of x will make a x2 + bx + c2 a complete square? 29. What value of b will make b2-4ac a complete square? 30. Find three square numbers which are in Arithmetic Progression. Ans. (m2 - n2 - 2mn)3, (m2 + n2)3, (m3 — n3 + 2mn)3. SCALES OF NOTATION. 1. 17486 is in the denary scale, find the equivalent number in the senary scale. Ans. 212542. 2. 215855 is in the denary scale, find the same number in the duodenary scale. Ans. t4tee. 3. 3t4e2 is in the duodenary scale, find the same number in the denary scale. Ans. 80198. 4. Transform 1534 from the senary to the denary scale. Ans. 418. Ans. 1456. Ans. 14tee. Ans. 565 ft. 8'. 7′′. 6′′′′. 8. The difference between any number (in the denary scale) and that formed by reversing the order of the digits is divisible by 9. Prove it. 9. Extract the square root of 25400544 in the senary scale. Ans. 4112. 10. Extract the square root of 32e75721 in the duodenary scale; and then verify the result by squaring it. 11. The number 124 in the denary is expressed by 147 in another scale, required the radix of the latter. Ans. 9. 12. In what scale of notation will a number that is double of 145 be expressed by the same digits? Ans. Radix = 15. 13. Find the scale to which 24065 belongs, its equivalent in the denary scale being 6221. Ans. Radix = 7. 14. The area of a rectangle is 29ft. Oin. 4'., and its length is 12ft. 8 in.; find its breadth. Ans. 2ft. 3 in. 6'. 15. The area of a rectangle is 971 ft. 10 in., and breadth 24ft. 9in.; find its length. Ans. 39 ft. 3in. 2'.3". 16. The area of a square is 17ft. 4 in. 6'., what is the length of the side? Ans. 4ft. 2in. 0'. 2′′. 10". 17. What is the cube of 6ft. 6in.? 18. Ans. 188ft. 1090 in. Prove that any number of 4 digits in the denary scale is divisible by 7, if the first and last digits be the same, and the digit in the place of hundreds be double that in the place of tens. 19. Any number is divisible by 4, if the last two digits, taken in order to form a number, be divisible by 4. 20. Any number is divisible by 8, if the number, consisting of the last three digits in order, be divisible by 8. 21. There is a certain number consisting of two digits, which is equal to four times the sum of its digits; and if to the number 18 be added, the digits will be reversed. Ans. 24. 22. There is a certain number, a multiple of 10, which exceeds the sum of its digits by 99; find the number. Ans. 100. 23. Prove that the sum of all the numbers which are composed of the same digits is divisible by the sum of the digits. 24. Find the greatest and the least numbers of 4 digits in the senary scale, as expressed in the denary scale. Ans. 1295, 216. 25. The two digits of which a number consists are as 3: 2, and when the digits are reversed the number is divisible by 3, and is to the former number as 23: 32. Required the number. Ans. 96. 26. Any number consisting of an even number of digits, in a system whose radix is r, is divisible by r + 1, if the digits equidistant from each end are the same. |