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& 14 TAE Marine. REVIEW SCIENTIFIC LAKE NAVIGATION. By Clarence E. Long. Find the least common multiple of 3, 9, 7, 14, 6, 14, 2, 12. 214 0.7% %4 16-14 2 +12 Ke) 7 6 3 7 2 3 is stricken out since it is a factor of 6, which 'is one of the numbers. 7 is a factor of 14, one I4 is stricken out. GO is a factor of 12. 2 is a-factor of 12. The least common multiple of the remaining numbers, 9, 14, and 12, is to be found. Divide these numbers by a prime number that is exactly contained in any two of them, bringing down the numbers that are not multiples of the divisor. Taking 2 as a divisor, bring down 9, and write quotient 7 and 6. 3 being a factor of two of the three numbers, 9, 7, 6, is taken as the next divisor. 3 is written as a quotient, 7 is brought down, 2 is a quotient. As there is no factor common to any two of the numbers, 3, 7, 2, we find the least common multiple by multiplying together the two divisors and these three numbers, thus: 2X 3%3 xX 7 X 2 = 252, least common multiple. Change 1-3, 3-9, 2-5, 7% and 8-9 to fractions having the least common denominator. 360 is the number, 1-3 = 120- 360; 3-9 = 120-360; 2-5 = 144-360; 74 = 315-360; 89. = 320-360. Examples for practice--Reduce to fractions having the least common denominator. 7%, 11-16 and 17-24. Ans. 42-48, 33-48, 34-48. 4-13, 15-26, 7-39. Ans. 24:78, 45-78, 14-78. 20-21, 9-56, 5-84. Ans, 160-168, 27-168, 10-168. 6%, 7-20, 7 and 1%. Ans. 125-20,.7-20, 140-20, - 30-20. When the prime factors of the given numbers cannot be. discovered easily by inspection they may be found. by the same method as that of finding the greatest common divisor, which will be explained later. Example.--Find the least common multiple {Of '255¢ and 357: Me Explanation.--Since the factors of 'the numbers © cannot be readily discovered by inspection, the greatest. common divisor is found to be 51, Dividing each of the given num- bers by 51, gives the quotients 5 and 7. which are prime to. each other. Therefore, 51 X 5 X 7 = 1,785, which is the least common multiple of 255 and 357.: g55.) 357° (11 255.5 102.) 255) '(42 204 SP.) 102.( /2> 102 © Therefore,.- 51 is the e C £), 51 ) 255 357 5 7 51 X§ X 7 = 1,785. COMMON DIVISORS. A number that is an exact divisor of two or more numbers is called a common divisor of the numbers. The greatest number that is an exact divisor of two or more numbers is called the greatest common divisor of the numbers. Principle--The greatest common divisor of two or more numbers is the product of all their common prime factors. Sometimes the numbers cannot be readily factored. In such cases the following method is employed: What is the greatest common divisor. of 35 and 168? a5.) te ( & Explanation--The greatest com- 140 mon. divisor .cannot be greater ee than the smaller number; there- Oa) 35° (1 fore 35 will be the greatest com- 28 mon divisor if it is exactly con- -- tained in 168... By trial it is 7) 28 (4 found that it is not an exact di- 28 visor of 168, since there is a re- -- mainder of 28. Therefore, 35 is not the greatest common divisor. Since 168 and 140, which is 4 times 35 are each divisible by the G. C. D.,, their difference, which is 28, inust be divisible by the greatest C. D.; therefore, the G. C. D., can- not be more than 28. Then 28 must be the G. C. D. if it will exactly go into 35. By trying we find that it does not, but leaves a remainder of 7, so 28 is not the G.C. D. We then see that the number cannot be any larger than 7, and if 7 is exactly contained in 28 it must bathe G. C. D. Rule.--Divide the greater number by the less, and if there is a remainder divide the less number by it, then if there is no remainder the last divisor is the greatest common divisor. If there is a remainder proceed as before by dividing the previous divisor by the last remainder until it comes out equal, when the last divisor will be the G. C. D. > Tf more than two numbers are given find the G. C. D. of any. two and then of the divisor and another of the numbers, and so on until all of the numbers are used. The last number will be the greatest common divisor. FRACTIONAL ADDITION. Fractions can be added only when they have a common denominator, and when they express parts of like units. A'common denominator is a number that will exactly con- tain all the denominators. The least common.denominator is the least number that will exactly contain all of the denominators. : What is the sum of 3 and 544? Ans..8-8, or 1 whole one. Sold 5-15 of a vessel to one man, 7-15 to another and 3-15 to another. How much was sold to all? Ans. 15-15, or the whole of it. » Mary paid $34 for. some ribbon, and $% for a pair of gloves. How much did she pay for both? Ans. $1. Rule for adding common fractions: Reduce the given fractions to equivalent fractions having the least common denominator, and write the sum of the numerators over the common.denominator. Reduce if pos- sible. Add together 4, % and %,or 4% t+YwtR=? First, reduce your denominators to the least common. de- nominator, that is, your least common denominator wants to be some number that each one of the above denominators will be contained in it an equal number of times without a remainder. When you have found this least common de- nominator see how many times the first denominator is therein contained and with this quotient multiply it by the first numerator, setting it down above the common denomin- ator with a line between them. Do the same with each of the other denominators and numerators, and then add all your numerators together and this sum place over the least common denominator, and this will be the sum or amount