Elementary arithmetic, with brief notices of its history... by Robert Potts.

14

IIINTS, RESULTS, ETC., FOR THE EXERCISES.
I. The first 8 questions may be answered from Art. 1. 9. Arts. 6, 7.
11I. The following are the greatest common divisors of the pairs of numbers:
5: 72: 3: 9: 1: 448: 2: 1: 1: 15: 21: 29: 37: 71: 37: 29766: 7: 11: 1:
73: 5.
III. The following are the greatest common measures of each of the groups:
28: 8: 27: 17: 3: 42: 49: 40: 47: 5.
IV. The following are the least common multiples of the successive sets of numbers:
36: 180: 140: 200: 1710: 323: 11803: 18648: 13986: 720: 13000:
68035275: 361053770: 806265999: 51493672.
60: 495: 144: 180: 24216: 10080: 199800: 1040300: 123600: 190256:
315315: 6126120: 498400: 1980: 35640: 1021020: 11X13x17X19: 15120:
23x32x5: 7X11X13X17X31X47: 7200: 5040: 1200.
V. 1. 2, 4, 7, 14, 28, arc the common measures; and the least common multiple is
1216, and the largest under 10,000 is 9728. 2. 2250. 3. 14549535. 4. 8640. 5.
2t4.33.5.72.112.13.17.113. 6. 1080, 1440, 1800. 7. Add 3 to the L.C.M. of the
given numbers. 8. The quotient arising from dividing the L.O. M. by the G.C.M.
of two numbers, consists of the product of two numbers prime to each other. The
quotient in this case is 315, which is to be divided into two such numbers: find the
numbers and verify the process. 9. Arts. 6, 7. 10. Any multiples greater than 6
of 143 will fulfil the conditions. 11. The G.C.M. is 33 57l11, from which all
the common measures may be found, which are 31. 12. The greatest number is 20.
VI. 1. Art. 3. 2. Omit the 21 prime numbers between 100 and 200, and 79 composite numbers remain, which are readily resolved into their prime factors. 3. Employ
the method of Art. 3. 4. Examine the series of prime numbers. 5. 24.33.52.7.11.
6. The numbers when resolved into their prime factors are 23.3'.7.13, and 24.3.5.17,
in which the common factors are 23.3. 7. 21. 8. The numbers when resolved are
2(s.3.13.53, 26.32.13.43, 23.3.13.443, and 26.32.13'2: of which the G.C.M. is 23.3.13,
and the L.C.M. is 26.32.131.43.53.443. Art. 5. 9. Resolve 1680 into its prime
factors. 10. See Art. 5. L.C.M. is 2.34.5.7.13.17. 11. See Art. 2. 12. By
means of the G.C.M. the numbers are found to be 29, 13, 37.
VII. 1. Take the prime numbers 13, 17, 31, 37, 41, 43, 61, 71, 211, 311. 2. Into
twenty several different ways. 3. See Art. 2. 4. See Art. 2. 5. See Art. 5 and the
note. 6. The truth of the statement may be verified by taking any other prime
numbers. 7. The multipliers are respectively 8547, 17094, 25641, 34188, 42735,
51282, 59829, 68376, 76923. 8. The following are the first periods respectively of
the ten quotients: 153846, 230769, 307692, 384615, 461538, 538461, 615384, 692307,
769230, 846153. The other two properties may be readily verified. 9. The numbers
are 11, 13, 17, 19, which can be found by means of the G.C.M. 10. 125, 225, 325.
VIII. 1. The least number of ounces of standard gold is 80. 2. ~21 can be paid
by 20 guineas, or by 168 half-crowns. 3. The least sum is ~2 3s. 10~d., whicl
can be paid by 13 roubles or 54 francs. 4. The dollar, the franc, and the shillinl
respectively consist of 2277, 414, and 528 elevenths of a shilling.
IX. 1. 724 grains Troy. 2. 361bs. Avoird. 3. 5554 ounces Avoird. 4. T]he
greatest weight is 40 grains. The least 175lbs. Troy, or 144 Avoird. 5. 45 gallons.
X. 1. 4 revolutions of the larger wheel are equal to 5 of the' smaller which caii
be made in running 60 feet. 2. 180 pebbles. 3. They stepped together 4440 times.
The man took 8800 steps, the woman 13320, and the boy 17600. 4. 12 feet long.
5. 360 sheep. 6. 47 and 59 units; the unit being 16. feet, or 198 inches.
XI. 1. 2674 days 9min. 59sec. 2. 23148 and 720720 seconds can be represented by
643 and 20020 when the unit of time is 36 seconds. 3. The greatest unit of time is
57 minutes. 4. In 2min. 48sec. 5. The interval will be 62370 seconds. The four
points will have moved over the distance 315, 125, 70, 54 times respectively. 6.
The three bodies will be in conjunction again after 7113120 days, when the first will
have made 81760 revolutions in its orbit; the second, 31755; and the third, 19488.