EXERCISE LXIV. EXAMPLES IN DIVISION OF INTEGERS AND DECIMALS BY A SINGLE DIGIT (OR BY 11 OR 12) IN ANY PLACE OF INTEGERS OR DECIMALS. ABSTRACT NUMBERS. 4. Of 129.8643 5. Of 64.9548 6. Of 1. 7. Of 7-875 8. Of 118 and 9. Of 00004 10 Of 40000 ⚫008 14. Divide 84 by 005 and by 500. 30000 15 Dive. 1. by 2 by 20 by 200 & by .02 ·80000 | 16. Divide 4 by 08 by 80 and by '008 00005 17. Divide 3 by 120 and by 4000 1108 18. Divide 7 by 08 by 800' and by 0008 00004 19 Divide 6 by 0012 and by 00008 20. Divide 5 by 02 by 04 and by 800. ⚫0004 DIVISION OF ANY INTEGER OR DECIMAL BY ANY INTEGER OR DECIMAL. Any Number whatever would, if we had Digits enough, be expressed by a single Digit; and that single Digit would occupy the right-hand place of the Digits which now express the number. (See page 115.) Therefore, any combination of Significant figures may be considered as equivalent to a possible single Digit in their right-hand place; just as 11 and 12 are always treated. It follows, then, that Principle XXI. is true, not only of Numbers expressed by single Digits, but of any Number whatever. EXTENSION OF PRINCIPLE XXI. Dividing by any Integer or Decimal removes every Quotient-figure as many places, BUT IN THE CONTRARY DIRECTION, as right-hand Significant figure of Divisor is from Units' place. Now, 295460395· 748, therefore the Natural Place of first Quotient-figure 7 is the third place of Integers. .. 29546039.5 295460 3.95 295460÷ *395 295460÷ *0395 &c. Also 295460÷ = to the 7480 Each Quo. Fig. being moved one place 74800 Each Quo. Fig. being moved two places left. 748000 Each Quo. Fig. being moved three places =7480000 Each Quo. Fig. being moved four places &c. &c. Why? &c. 395074.8 Each Quo. Fig. being moved one place 295460 + 39500 7:48 Each Quo. Fig. being moved two places 295460395000 = 748 Each Quo. Fig. being moved three places 2954603950000= 0748 Each Quo. Fig. being moved four places Again, 29546 ÷ 395. 74.8, where Natural Place of first Quotient-figure 7 is the second place of Integers. To Divide any Integer or Decimal by any Integer or Decimal. First. Decide the Natural Place of First Quotientfigure. Second. Divide as usual; but set every Quotient-figure as many places to the LEFT or RIGHT of its Natural Place, as right-hand Significant Figure of Divisor is to the RIGHT OF LEFT of Units' place. Observe. If Divisor be an Integer with one or more ciphers adjoining Decimal Point, consider these as no part of the Divisor. Their only effect will be to remove the Quotient-figures so many places to the right. EXERCISE LXV. EXAMPLES IN DIVISION OF ANY INTEGER OR DECIMAL BY ANY INTEGER OR DECIMAL. ABSTRACT NUMBERS. 12. Of, 566319.6 3.96 6. Of 36921335738-4267 13. Divide 69338.51 by 97 by 97 and by '0097 14. Divide 1 by 512 by 5120 by 5·12 by 512000 and by ⚫512 15. Divide 7 by 40.96 by 04096 by 512 by 000512 and by 1024000. 16. Divide 16.5 by 256, and 29.875 by 064 17. By what Number must I multiply 3.125 to produce 2281250· ? GREATEST COMMON MEASURE. A Measure of any Number is an exact Divisor of it; that is to say, one which leaves no Remainder. 1262, 18· ÷ 6′ — 3'; no Remainder in either instance. = = Therefore, 6 is a Measure of 12, and of 18'. 1562, 206: 32; Remainders 3 and 2.. Therefore, 6 is not a Measure of 16, or of 20. Two things are, then, evident. First. "" Measure is only another name for "Factor." Second. Any Number is a Measure of its own Multiples only. A Common Measure of any two or more Numbers is a Factor of each of them. The Greatest Common Measure of two or more Numbers is the greatest Number which is an exact Divisor (or Factor) of each of them. For the words, “Greatest Common Measure," we write G.C.M. 3. is a Common Measure of 12., 18., and 42·. Find by Inspection, EXERCISE LXVI. 1. A Common Measure of 16., 24., 18., 32, and 100. 2. Two Common Measures of 72, 18, 24, 108, and 30. 3. Three Common Measures of 96, 144, 18, 48, 30, and 36.. 4 Four Common Measures of 30, 120, 450, and 900.. 5. The G.C.M. of 28, 42, and 56; and G.C.M. of 30, 24, 18, and 12.. 6 The G.C.M. of 36, 45, 72, and 27; and G.C.M. of 22, 143', and 187.. 7 The G.C.M. of 24, 16, 56, 112, and 64'; and G.C.M. of 45', 78', and 91.. 8. How many Common Measures have 16, 24, 36', and 48.? 9. What Numbers less than 100 are measured both by 4 and by 6⚫? 10 Name the first ten Nos. of which 6, 4, and 5 are Common Measures. GREATEST COMMON MEASURE OF ANY TWO NUMBERS. In ordinary Arithmetic we never have occasion to find the G.C.M. of more than two Numbers at a time. The method of finding the G.c.M. of any two Numbers is derived from four simple facts. I. Any Number is a Measure of its own Multiples only (See page 130.). II. The Sum of two Multiples of a Number is also a Multiple of that Number. For, considering, for example, 8 as a Complex Unit, it is plain that, ··5 +7. = 12... 5. (eights) +7 (eights) = 12. (eights.) .. 5th Multiple of 8+ 7th Multiple of 8. 12th Multiple of 8', and so on with any two Multiples of any other Number. III. THE DIFFERENCE of two Multiples of a Number is also a Multiple of that Number. For, 75. = 2. .. 7. (eights) 5. (eights) 2. (eights). .. 7th Multiple of 8 5th Multiple of 8 2nd Multe. of 8', and so on with any two Multiples of any other Number. IV. ANY MULTIPLE of a Multiple of a Number is also a Multiple of that Number. = 14. (eights.) For, 7 X 2. 14. .. 7 (eights) × 2· = 14th Multiple of 8. These four facts, combined, constitute PRINCIPLE XXII. bers measures also a. THEIR SUM. Every Measure of any two Num b. THEIR DIFFERENCE. c. ANY MULTIPLE OF EITHER OF THEM. The process for finding the G.C.M. of two Numbers is exhibited in the following Example worked out. Find the G.C.M. of 1387. and 2508. To prove that this method is the true one, we have to shew two things. First. That 19 must be a Common Measure of 1387. and 2508.. Second. That there is no Common Measure of those two Numbers, greater than 19'. For the first part of the proof, we trace the work upwards from 19.. 19 is a Measure of 38. For 38.÷ 19. leaves no Remainder. For 57= 19 is, then, a Common Measure of 2508 and 1387. For the second part of the proof, we trace the work downwards to 19. Now, if 19. be not the G.C.M. of 1387 and 2508', some Number greater than 19 is their G.C.M. Suppose that a greater Number than 19 measures 1387 and 2508; then, and 2508 . A greater No. than 19 measures 1387 For 11212508-1387 PR. XXII. b. But this conclusion is absurd. Consequently, the supposition that led us to it is false. There is, then, no greater Common Measure of 1387 and 2508 than 19. Hence, 19 is the Greatest Common Measure of 1387. and 2508, and the process is proved to be the true one. Numbers prime to each other. Numbers which have no Common Measure are prime to each other, although neither of them may be a really Prime Number. (See page 43.) Thus, although 8°, 15°, 49', and 187 are all Composite Numbers, they are prime to each other, for they have no Common Measure. We shall call Numbers which are prime to each other, "Primes." THE LOWEST TERMS OF A FRACTION. We have already seen that the same Fraction may be expressed by different Numbers; for, by PRINCIPLE XIII., &c. &c. or or or or 12 or 14 or 2 or 1500 When a Fraction has its lowest possible Numerator and Denominator, we say it is in its lowest terms. = Thus, 2, in its lowest terms, is written. See above. Now, since Dividing any Number by 1 makes no altera tion in it, and since is equal to 1'; therefore, = ; but, ÷ 3 = 11⁄2. See third method of Dividing Fraction by Fraction, p. 123. M |