NOTES.-1. If one number is contained in any of the others, it may be omitted.

2. If the number cut off is found to be prime to the others in the same line, cut off another and proceed as before, reserving the first as a factor of the L. C. M.

Find the least common multiple of

2. 30, 80, 120, and 135.

3. 77, 91, 143, and 165.

Ans. 2160.

Ans. 15015. Ans. 237336.

4. 93, 132, 232, and 319.

GENERAL PRINCIPLES

OF GREATEST COMMON DIVISOR AND LEAST COMMON MULTIPI.E.

189. These General Principles express the relations between the greatest common divisor and the least common multiple.

1. The greatest common divisor of two or more numbers is a divisor of their least common multiple.

2. The product of two numbers divided by their greatest common divisor equals their least common multiple.

3. The product of the relatively prime parts of two or more numbers multiplied by the G. C. D. equals the L. C. M.

4. The quotient of the L. C. M. of two or more numbers, divided by their G. C. D., equals the product of the factors not common.

5. The prime factors not common may be found by resolving the quotient of the L. C. M. divided by the G. C. D. into its prime factors.

6. The G. C. D., multiplied by each of the factors not common, will give numbers having the same G. C. D. and L. C. M.

NOTE. The pupil may be required to illustrate these principles.

PRACTICAL PROBLEMS.

1. The L. C. M. of 6 and 8 and a number prime to each of them is 120; what is the third number?

SOLUTION.-120 contains all the factors of 6, of 8, and of the 3d number; hence all the factors of 120 not found in 6 and 8 constitute the third number. The only factor is 5, therefore 5 is the uumber required.

OPERATION.

120 23x3x5
6=2x3
8=2×2×2

.. 5 the number

2. The L. C. M. of 8, 12, and 45, and another number prime to each, is 2520; required the number.

Ans. 7.

3. The G. C. D. of two numbers is 5, and their L. C. M. is 30; what are the numbers? Ans. 10, 15, or 5 and 30.

4. The L. C. M. of 6, 9, 10, and a fourth number, is 630; what is the smallest number that it may be?

5. The G. C. D. of two numbers is 12, and is 72; required the numbers.

Ans. 7. their L. C. M

Ans. 24, 36.

6. The G. C. D. of three numbers of two factors each is 7, and their L. C. M. is 210; required the numbers.

Ans. 14, 21, 35.

7. What two numbers between 13 and 78 have the latter for their L. C. M. and the former for their G. C. D.?

Ans. 26, 39.

8. What three numbers of two factors each between 17 and 510 have the former for their G. C. D. and the latter for their L. C. M.? Ans. 34, 51, 85.

9. Find a number between 209 and 247 which has with each of them the same G. C. D. that they have with each other. Ans. 228.

10. Find 3 numbers between 161 and 1265 which have the same L. C. M. as these numbers.

Ans. 253, 385, 805.

11. Find 3 numbers between 119 and 187 which have the same G. C. D. as these numbers. Ans. 136, 153, 170.

12. Required three numbers between 119 and 374 which have with these numbers the same L. C. M. as the numbers themselves. Ans. 154, 187, 238.

13. The G. C. D. of four composite numbers of two factors each is 11, and their L. C. M. is 2310; what are the numbers? Ans. 22, 33, 55, 77 14. Required all the numbers whose G. C. D. is 45 and L. C M. is 4680.

Ans. 45, 90, 180, 360, 585, 1170, 2340, 4680

1824

Divid 021

1501

110

9. Divide 432 × 529 × 441 by 27 × 23× 7× 9.

Ans. 2576.

10. Divide 9801 × 2025 × 2401 by 891× 45 × 77.

PRACTICAL PROBLEMS.

Ans. 15435.

1. How many yards of alpaca, at 48 cents a yard, can be obtained for 36 bushels of corn at 84 cents a bushel?

SOLUTION.-If one bushel of corn is worth 84 cents, 36 bushels are worth 36× 84 cents; for 36×84 cents at 48 cents a yard, we can get as many yards of alpaca as 48 is contained times in 36×84, which we find by cancellation to be 63.

OPERATION.
3 21
36×84

48

=63 Ans.

2. How many barrels of pork, at $16 a barrel, can be obtained for 64 tons of hay, at $23 a ton?

Ans. 92. 3. A merchant sold 18 hhd. of molasses, each containing 75 gal., at 64 cents a gal., and received in payment a number of chests of tea, each containing 24 pounds, at 90 cents a pound; how many chests were there? Ans. 40.

4. Multiply 45 by 6 times 25 and divide by 91; multiply the quotient by 13 times 63 and divide by 81; multiply this result by 12 times 19 and divide by 6 times 95. Ans. 300.

5. A dealer exchanged Minnesota extra flour, at $9.50 per barrel, for 19 cases of children's shoes, each containing 60 pairs, at $1.25 a pair; how many barrels of flour were exchanged? Ans. 150 barrels.

6. A commission merchant sold 21 bales of "middling upland" cotton, each containing 400 pounds, at 16 cents a pound, and received in payment 16 hogsheads of molasses, containing 120 gallons each; what was the cost of the molasses per gallon? Ans. 70 cents.

7. A grocer bought 7 chests of souchong tea, containing 24 pounds each, at $1.05 per pound; how many firkins of butter, at 35 cents a pound, will be required to pay for it, each firkin containing 56 pounds? Ans. 9 firkins.

6

SECTION IV.

COMMON FRACTIONS.

192. A Fraction is a number of the equal parts of a unit; as 3 fourths.

193. A Fractional Unit is one of the equal parts of the Unit. A Fraction is a number of fractional units.

194. Similar Fractional Units are those which are alike; as 2 fourths, 3 fourths.

195. Dissimilar Fractional Units are those which are unlike; as 3 fourths, 4 fifths.

196. Fractions are divided into two classes; common fractions and decimal fractions.

197. A Common Fraction is one in which the unit is livided into any number of equal parts.

198. A Decimal Fraction is a number of the decimal divisions of the unit.

units.

NOTES.-1. Units are distinguished as Integral units and Fractional The word Unit, without any qualifying word, means the Integral unit. When the term fraction is used without any qualifying word, the common fraction is meant.

2. A fraction implies three things: 1st, a thing to be divided; 2d, equal parts of the thing; and 3d, the number of parts taken-that is, the integral unit, the fractional unit and its relation to the integral unit, and the number of fractional units taken.

3. The primary conception of a fraction is that it is a number of equal parts of a unit. It may, however, be regarded as a number of equal parts of one thing, or one equal part of a number of things. Thus, four fifths may be regarded as four-fifths of one or one-fifth of four.

199. A Common Fraction is expressed by two numbers, one written above the other, with a line between them. Thus, expresses 4 fifths.

200. The Denominator denotes the number of equal parts into which the unit is divided; it is written below the line.

201. The Numerator denotes the number of equal parts which are taken; it is written above the line.