8. What is the least common multiple of 6, 8, 12, 18, 24, 131 and 137? Ans. 1292184. 9. What is the least common multiple of 8, 15, 77 and 221? 10. What is the least common multiple of 10, 15, 45, 75 and 90? (b) It is evident that 10, 15 and 45, in the above example, may at once be struck out; for each of these numbers is a measure of 90, and .. whatever multiple of 75 and 90 is found, it, certainly, must be a multiple of 10, 15 and 45; hence the question is reduced to this: What is the least common multiple of 75 and 90? NOTE.-Many other abbreviations of this and other rules may be effected, but a delicate perception of the relations of numbers, and a skilful application of principles, will much more facilitate the progress of the learner than any set of formal rules. 11. What is the least common multiple of 4, 9, 6 and 8? Ans. 72. 12. What is the least common multiple of 8, 12, 16, 24, 32, 48 and 96? Ans. 96. 13. Find the least common multiple of 80, 20, 160, 40, 5, 320, 10 and 16. 14. Find the least common multiple of 91, 3523 and 6487. Ans. 12305839. 15. Find the least common multiple of 12089, 1309, 2849 and 2233. 16. Find the least common multiple of 28, 42, 56, 70, 80 and 90. (c) If the numbers are prime, or even mutually prime, their product is their least common multiple. 17. What is the least common multiple of 8, 15 and 77? Ans. 9240. 18. Find the least common multiple of 1181, 2741 and 3413. (d) The least common multiple of two numbers is equal to their product divided by their greatest common measure. 19. What is the least common multiple of 12 and 20? The greatest measure of 12 and 20 is 4, and 20. What is the least common multiple of 35 and 49? 21. What is the least common multiple of 39 and 91? 115. The following facts will be found convenient in the subsequent rules: (a). Every number whose unit figure is 0, or an even number, is itself even, and .. divisible by 2. (b) If the two right hand figures of a number are divisible by 4, the whole number is also divisible by 4; e. g. 8724 : 8700 +24; now, 100 is divisible by 4, and .. 8700 is divisible by 4 (111, b); again, since 8700 and 24 is each divisible by 4, their sum, 8700+24= 8724, will also be divisible by 4 (111, c). (c) Similar reasoning will show that the whole of a number is divisible by 8, if its last three figures are divisible by 8, etc., etc. (d) A number ending with 5 or 0 is divisible by 5. (e) Any number ending with 0 is divisible by 10. (f) Every number the sum of whose digits is divisible by 9 is itself divisible by 9; e. g. 5643 = 5000 + 600 + 40 + 3. = Now 5000 5 X 1000 5 X (999 + 1) = 5 × 999 +5, 6006 X 100 40 = 4 × 10 = 6 X (991) = 4X (9 + 1) = .. 5643 = 5 × 999 + 6 × 99 +4 × 9 + 5 X 6 X 996, 4 × 9 +4; + 6 + 4 + 3; again, it is evident that 5 × 999 + 6 × 99 + 4 × 9 is divisible by 9, and if 5 + 6 + 4 + 3 (= 18) is divisible by 9, then the whole number, 5643, must be divisible by 9; but it will be seen that 56 + 4 + 3 is the sum of the digits which express the number 5643; hence any number is divisible by 9 if the sum of its digits is divisible by 9. If the sum of the digits of a number divided by 9 give a remainder, then the number itself divided by 9 will give the SAME remainder. NOTE. — It is on these properties of the number 9 that the rules often given, for proving Addition, Subtraction, Multiplication and Division by casting out the 9's are founded. (g) The properties given for 9 are equally true for 3; i. e. if the sum of the digits of a number is divisible by 3, the number is itself divisible by 3, and if the sum of the digits divided by 3 gives a remainder, then the number divided by 3 will give the same remainder. (h) Any even number divisible by 3 is also divisible by 6; for, since it is even, it is divisible by 2, and, being divisible by 2 and by 3, it is divisible by 2 × 3 = 6. NOTE. The properties named in the foregoing paragraphs are dependent upon the given conditions; e. g. in (a), a number not ending in 0 or an even number is not divisible by 2; etc. (i) Every prime number, except 2 and 5, must end with 1, 3, 7 or 9; for, 1st. Every number must end with some one of the ten digits, 2d. But no even number, except 2, is prime; .. take away 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; 2, 4, 6, 8, NOTE. The converse of this proposition is not true; i. e. a number ending with 1, 3, 7 or 9 is not necessarily prime. § 10. VULGAR FRACTIONS. 116. A FRACTION* is an expression representing one or more of the equal parts into which a unit is supposed to be divided. 117. A Vulgar or Common Fraction is expressed by two numbers, one above and the other below a line; thus, (one half), (two fifths), etc. (a) The number below the line shows into how many parts the unit is divided, and is called the denominator, because it denominates or gives name to the parts; thus, if a unit is divided into 3 equal parts, each part is one third; if into 8, each part is one eighth; etc. (b) The number above the line is called the numerator, because it numerates or numbers the parts taken. (c) The numerator and the denominator are the terms of the fraction. 118. A fraction is nothing more nor less than unexecuted division, i. e. division indicated but not performed, the numerator being the dividend and the denominator the divisor. This is the key to a knowledge of fractions; and this knowledge of fractions is, in turn, the key to Higher Arithmetic and Algebra. He who has the key intelligently in his possession will advance rapidly and pleasantly, while he who neglects the key will see no beauties in mathematics. (a) It follows from the above, that the value of a fraction is the quotient of the numerator divided by the denominator; thus, 12 = 12 ÷ 4 - 3. 119. A proper fraction is one whose numerator is less than the denominator; as, 3, 24, etc. 8 120. An improper fraction is one whose numerator equals or exceeds its denominator; as, 4, 7, 8, 15, etc. An improper * Fraction, from the Latin frango, to break. fraction equals or exceeds a unit; hence its name fraction. 121. A simple fraction has but one numerator and one denominator, and is either proper or improper; as, 3, 3, 8, 42, etc. 122. A compound fraction is a fraction of a fraction; as, of, of of §, etc. 123. A mixed number is a whole number and a fraction united; as, 34, 203, etc. 124. A complex fraction is one that has a fraction or a mixed number for one or for each of its terms; as, 31237 7'6' 2'2' 81 etc. 7228131 REMARK. The following are the most important operations in fractions. CASE 1. 125. To multiply a fraction by a whole number, RULE 1.-Multiply the numerator by the whole number (59, a, and 118); or, RULE 2. - Divide the denominator by the whole number (59, d). Ex. 1. Multiply by 3. NOTE 1. X3, by Rule 1; or, – The 2d rule is preferable in this and all similar examples, because it gives the result in smaller terms. |