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sor. We may therefore cast out the factor 8 from 728. The other factor is 91; therefore, the G. C. D. required is the same as the G. C. D. of 91 and 1183, which is 91.

The work, then, would be written thus:-
:-

4277) 9737 (2

1183) 4277 (3

7288 X 91 ) 1183 ( 13

273

00

Hence, 91 = = G. C. D.

(f) What is the greatest common divisor of –

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115. Common Multiples, Definitions, and Properties.

(a.) A multiple of a number is any entire number which can be exactly divided by it.

(b.) A common multiple of two or more numbers is a number which is a multiple of all of them.

(c.) The least common multiple of two or more numbers is the smallest number which is a multiple of all of them.

(d.) From these definitions and the principles previously established, it follows that,

1. A multiple of a number must contain all the prime fuctors of that number.

2. A common multiple of two or more numbers must contain all the prime factors of each of them.

3. The least common multiple of two or more numbers must be the smallest number which contains all the prime factors of each of them.

116. Least Common Multiple.

Method by Factors.

1. What is the L. C. M.* of 504, 756, 924, and 1176?

* L. C. M. means least common multiple.

Solution. - The L. C. M. of these numbers is the smallest number which contains all the prime factors of each of them.

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The factors of 1176 are 23 × 3 × 72, all of which we take. The fac tors of 924 are 22 X 3 X 7 X 11, all of which, except 11, we have taken. We therefore introduce the factor 11, which gives 23 X 3 X 72 X 11. The factors of 756 are 22 X 33 X 7, all of which, except 32, we have taken. We therefore introduce the factor 32, which gives 23 X 38 X 72 X 11. The factors of 504 are 23 X 32 X 7, all of which we have. Therefore, the product of 23 × 33 × 72 × 11, which is 116424, is the L. C. M. required.

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NOTE. - When the factors of the L. C. M. have been obtained, much labor in multiplying may be saved, by writing some one of the numbers in place of its factors. Thus, in the example above, by writing 1176 instead of its factors, we should have 1176 X 32 X 11 = 116424 = L. C. M., as before.

What is the least common multiple 2. Of 18, 24, and 42?

3. Of 36, 48, and 60?

4. Of 132, 144, and 160?

5. Of 98, 126, and 186?

6. Of 364, 637, and 1547 ?

7. Of 605, 325, 715, and 1001 ?
8. Of 504, 756, 1008, and 1512?

9. Of 594, 1386, and 1782 ?
10. Of 735, 945, 1365, and 2310?
11. Of 154, 231, 264, and 392 ?

117. Abbreviated Method.

(a.) When the factors of the several numbers can readily be determined, or after they are found, much labor may often be avoided by considering what factors are wanted with any one of the numbers to produce the L. C. M.

(b.) Thus, in the example solved in 116, since the factors cannot easily be recognized, we write them thus:

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(Then, since 1176 contains all the factors of each of the other numbers, except 11 X 32, the L. C. M. must equal 1176 × 11 × 32; or if divided by 1176, the quotient would be 11 X 32, or 99.

(d.) Again. Since 924 contains all the factors of each of the other numbers, except 2 × 7 × 32, the L. C. M. must equal 924 × 2 × 7 X 32; or, divided by 924, the quotient must be 2 X 7 X 32, or 126.

(e.) Similar considerations will enable us to obtain the L. C. M. from 756 and from 504. A little practice will enable the pupil to determine at a glance from which number the L. C. M. can best be obtained.

1. What is the L. C. M. of 24 and 36?

Solution. Observing that 36

3 X 12, and 24 = 2 × 12, it will

at once be seen that the L. C. M. may be found by multiplying 36 by 2, or 24 by 3, and is 72.

2. What is the L. C. M. of 21, 35, and 56?

Solution. It is obvious that 56 contains all the factors of 21 and 35, except 3 and 5.* Hence the L. C. M.: = 56 X 5 X 3 = 840.

Second Solution.

- Since 21 contains all the factors of 35 and 56, except 5 and 8, the L. C. M. must be 21 X 5 X 8 = 21 X 40 = 840.

NOTE. In the above example, it is better to begin with 21 than 56, because it is easier to multiply 21 by 8 X 5, or 40, than to multiply 56 by 3 X 5, or 15.

3. What is the L. C. M. of 8 and 9?

Solution.- Since 8 and 9 have no common factors, their L. C. M. must be their product, which is 72.

(f.) What is the least common multiple

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15. What is the L. C. M. of 15, 18, 30, 12, and 36?

* For 56 = 8 × 7, 35 = 5 × 7, and 21 = 3 × 7; and 8, 5, and 3 are prime to each other.

Solution.

- Since 36 is a multiple of 18 and 12, and 30 is a multiple of 15, the L. C. M. required must be the L. C. M. of 30 and 36, which equals 5 X 36, or 6 X 30 180.

(g.) What is the least common multiple

16. Of 3, 6, 12, and 18?

17. Of 4, 9, 12, and 18?

18. Of 5, 3, 6, and 15?
19. Of 12, 24, 36, and 72?

20. Of 14, 15, 18, 30, 36, and 42 ?
21. Of 9, 10, 11, 12, and 18?
22. Of 98, 154, 198, and 284 ?
23. Of 72, 108, 180, and 252 ?
24. Of 306, 408, 612, and 1020?
25. Of 1548, 2322, 2580, and 3870?
26. Of 1872, 4212, 6318, and 8424?

118. When Factors cannot easily be found.

(a.) Since two numbers can have no other common factors than those of their greatest common divisor, it follows that the least common multiple of any two numbers may be found by multiplying one of them by the quotient obtained by dividing the other by their greatest common divisor. When prime to each other, their L. C. M. will be their product.

(b.) This principle may be advantageously applied when we wish to find the least common multiple of numbers the factors of which cannot easily be found.

1. What is the L. C. M. of 7379 and 9263 ?

Solution. As these numbers cannot readily be divided into their factors, we first find their G. C. D., which is 157. Dividing 7379 by 157 gives 47 for a quotient, which must contain all the factors of the L.C.M. that are not found in 9263. Hence the L.C.M. must equal 9263 × 47 = 435361.

(c.) What is the least common multiple

2. Of 5207 and 7493?

5. Of 3901 and 9047?

3. Of 2993 and 8651?

6. Of 6659 and 8083?

4. Of 3337 and 7471?

7. Of 7379 and 9263 ?

8. What is the L. C. M. of 3763, 5183, and 7261 ?

-

Suggestion. First find the L. C. M. of any two of the numbers, as 3763 and 5183, and then the L. C. M. of this result and the remaining number.

(d.) What is the least common multiple -
9. Of 2881, 4171, and 9313?
10. Of 2419, 4661, and 5609 ?
11. Of 3811, 6031, 7519, and 7661?
12. Of 8381, 9809, 7361, and 6817?

SECTION X.

FRACTIONS.

119. Introductory.

(a.) Ir an apple, an orange, a line, or any other thing, or any number, be divided into two equal parts, the parts are called HALVES of the apple, orange, line, or of whatever may have been thus divided. If two or any number of things of the same kind are each of them divided into two equal parts, the parts will, as before, be called halves of a thing of that kind, and there will be as many times two halves as there are things divided.

(b.) If any thing should be divided into three equal parts, the parts would be called THIRDS of the thing. If the thing divided should be an apple, the parts would be thirds of an apple. If each of several apples should be divided into three equal parts, all the parts, or any portion of them, would still be called thirds of an apple. Again; if I should cut out of an apple such a part as would be obtained by dividing it into three equal parts, and then cut from another apple such another part, these parts would still be called thirds of an apple, although one of them came from one apple and one from another. And so, however many or few such parts we may take from the same apple, or different apples, they would all be called thirds of an apple.

(c.) Hence, to have a third or thirds of any thing, quantity, or number, it is only necessary to have one or more such parts as would be obtained by dividing the thing, quantity, or number into three equal parts.

(d.) Such considerations, extended, give the definitions of the fol lowing article.

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