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PART II. LOWEST COMMON MULTIPLE 85. Multiples. If one number is exactly divisible by another, the first is called a multiple of the second.
Thus, 12 is a multiple of 4. Likewise, in algebra, x2y2 is a multiple of x; it is also a multiple of y and of xy. Similarly, a2–4 b2 is a multiple of a-2 b (see Formula VII) and of a+2 b.
86. Common Multiples. In arithmetic a number which is exactly divisible by two or more other' numbers is called a common multiple of them.
Thus, 48 is a common multiple of 6 and 12. Likewise, in algebra, an expression which is exactly divisible by two or more other expressions is called a common multiple of them.
Thus, 5 xy2 is a common multiple of x and y; it is also a common multiple of 5 and x. Similarly, q2—62 is a common multiple of a-b and a+b.
Again, 3 x2yz3 is a common multiple of x, y, and z.
87. Lowest Common Multiple (L. C. M.). The lowest multiple common to two or more numbers is called in arithmetic their lowest (or least) common multiple. It contains fewer prime factors than any other common multiple that the numbers can have.
Thus, of all the multiples common to 2 and 3 (such for example as 6, 12, 18, 24, 30, etc.) the lowest is 6; that is, 6 contains only the two prime factors 2 and 3, while all the other multiples in the list contain more. Likewise, the lowest common multiple of 12 and 18 is 36. (Why?)
Similarly, in algebra, we say that the lowest common multiple of two (or more) expressions is that multiple of them which contains the fewest possible prime factors. It is usually denoted by the abbreviation L. C. M.
Thus, the L. C. M. of 4 a and alb is 4 a2b. Likewise, the L. C. M. of a and a-4 is a(a-4), or a?–4 a. Again, the L. C. M. of the three expressions aéb, azy and x2yz2 is a-bx?yz2.
88. Important Property of the L. C. M. It is to be carefully observed that the L. C. M. of numbers or expressions must be exactly divisible by each of them, as follows from $ 87. For example, 12, which is the L. C. M. of 2, 3, 4, and 6, contains 2 six times, 3 four times, etc. Likewise, the L. C. M. of ab and 5 bạc is 5 ab?c, and this contains both ab and 5 boc exactly. Explain this.
89. To Find the L. C. M. The way in which the L. C. M. of several numbers is found is illustrated below.
EXAMPLE 1. Find the L. C. M. of 24, 36, and 60.
36=2 · 2 · 3 · 3,
60=2 · 2 · 3 · 5. The L. C. M. is therefore 184.108.40.206.5, or 360, since this contains all the factors of each of the three given numbers, and it is the least number that does so..
Similarly, the L. C. M. of several expressions in algebra is found in the manner illustrated below.
EXAMPLE 2. Find the L. C. M. of 10 a2b, 16 a2b3, and 20 a'b4. SOLUTION. We write 10 ab=2 · 5 · a. b,
16 a2b3= 2 · 2 · 2 · 2 · (263,
20 a3b4=2 · 2 · 5 · 93 · 64. The L. C. M. is thus seen to be 220.127.116.11.5. a3b4, or 80 a3b4, since this contains all the factors of each of the three given expressions, and at the same time it is made up of fewer factors than any other similar expression that can be found.
EXAMPLE 3. Find the L. C. M. of a4 — 10 a2 +9 and a? — 4 a +3. Solution. a*— 10 a?+9= (a2–9)(a2-1)=(a-3)(a+3)(a-1)(a+1),
State the L. C. M. of the expressions in each of the following exercises. 1. x2y3 and xy.
9. a(b+c) and a?. 2. 4 ab? and 6 ab.
10. (a+b)2 and a+b. 3. 6 a2c2 and 9 ac?
11. x’yz, xy®z, and x-y-z. 4. 8 x2 and 4 xy.
12. ab, bc, and cd. 5. 5 ay: and 10 aʼy.
13. 3 xy, 6 yz, and 8 xyz. 6. xyz and xyw.
14. 5 ab?, 10 bc2, and 4 abc. 7. a2bc2 and ab2c3.
15. 12 axạy, 4 xy?, and 9a-xy. 8. 12 a2 and 6 b2.
90. General Rule for L. C. M. From what we have seen in § 89 concerning the L. C. M., we may now state the following general rule.
RULE FOR FINDING THE L. C. M. OF Two OR MORE EXPRESSIONS.
Resolve each expression into its simplest factors.
Find the product of all the different factors, taking each factor the greatest number of times it occurs in any of the given expressions.
This product is the required L. C. M. of the given expressions.
WRITTEN EXERCISES Find the L. C. M. of the expressions in each of the following exercises. . 1. ab, ab, and abo. 5. x+y and ax+ay.
2. xy2, x3y3, and xy2. 6. x2+xy and x+y.
9. x2 - y2 and x2 + y2. 10. x2+2 x and x2 – 4. 11. 3 y+2 and 9 y+6. 12. x-1, x+1, and x2 – 1. 13. a? — 1 and a-a-2. 14. a? — 3 a+2 and a-2. 15. (x - y): and x2 – 2 xy+y. 16. a?+5 a+6 and a2 +7 a+12. 17. a?+4 a+3 and a2 +3 a+2. 18. 3 p2 +15 r+18 and 72 +6r+8. 19. a4–1, a2 – 1, and a-1. 20. 1–x?, x-1, and (x - 1)2. 21. x2+xy+xz+yz and x2 +2 xy+ya. 22. (x+y)3 and x2 - y2. 23. a?+4 a+4, a?–4, 4-a’, and a4 — 16. 24. x2+x-42, «2 – 11 x+30, and x2+2 x–35.
25. a? — 7 a+10, a? — 10 a+16, and a2–5 a+6. * 26. a3 — 53 and a? — 62. 27. a3 +63 and a? — 62.
28. 8 63 — C3, 4 62 —4 bc+c, and 4b-2 c. * 29. a3b3 – 27 and a2b2 — ab - 6. * 30. ax3 — a ́y3 and ax? — 2 azy+ay.
91. Algebraic Fractions. Algebraic fractions are like arithmetic fractions. Just as means 3=5, so (or a/b) means a:b. In the same way, "To means (a+5) -6.
6 The dividend is called the numerator, and the divisor is called the denominator. The numerator and denominator taken together are called the terms of the fraction.
The fraction a/b is read a divided by b, or a over b.
The student will soon see that fractions in algebra are subject to the same rules that govern fractions in arithmetic. They are reduced to lower or higher terms, added, subtracted, multiplied, and divided just as arithmetical fractions.
92. Principle. In arithmetic we often change the form of a fraction without changing its value. Such changes all depend upon the following principle.
PRINCIPLE. The numerator and denominator of a fraction may be multiplied or divided by the same number without changing the value of the fraction.
Thus, 3-3.2-6. Likewise,
a+b=_(+6) - 1