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Find the least common multiple of

2. 28, 32 and 64. 5. 12, 16, 18 and 24.
3. 36, 72 and 144. 6. 15, 20, 25 and 30.
4. 45, 70 and 90.

7. 18, 54, 90 and 180.

What is the least common multiple of

8. 22, 55 and 77 ? 13. 12, 18, 24 and 96 ? 9. 48, 60 and 180? 14. 32, 56, 64 and 80 ? 10. 10, 64 and 96? 15. 14, 35, 50 and 28? 11. 81, 63 and 135 ? 16. 33, 99, 84 and 135? 12. 25, 70 and 95? 17. 17, 51, 65 and 121?

18. Find the least common multiple of 16, 20, and 30. 2D PROCESS.

ANALYSIS.—Since 2 is a prime fac2 116 20 30

tor of each of the numbers, it is also a

factor of the least common multiple. 2 8 10 15

(Prin.) Dividing, there remain as 5 4 5 15 the other factors of the numbers, 8, 4 1 3

10, and 15. 2 is a prime factor of 8

and 10, and is therefore a factor of 2 X 2 X 5 X 4 X 3 = 240 the least common multiple. Divid

ing, there remain 4, 5, and 15. 5 is a prime factor of 5 and 15, and is therefore another factor of the least common multiple. Dividing, there remain 4, 1, and 3, which are prime to each other. Therefore the product of the factors 2, 2, 5, 4, and 3, will be the least common multiple.

RULE.— Write the given numbers in a horizontal line. Divide by any prime number that is an exact divisor of two or more of the given numbers, and write the quotients and undivided numbers in a line beneath.

Thus continue to divide until the quotients and undivided numbers are prime to each other. The product of the divisors, and the numbers in the last horizontal line, will be the least common multiple.

In finding the least common multiple, all numbers that are factors of other given numbers may be disregarded. Thus, the multiples of 8, 16, 32, 64, 80, and 128 are the same as the multiples of 80 and 128.

EXAMPLES.

Find the least common multiple of
19. 60, 40, 120 and 72. 22, 32, 36, 72 and 80.
20. 81, 45, 108 and 135. 23. 30, 75, 60 and 90.
21, 40, 60, 80 and 120. 24. 24, 44, 65 and 100.

What is the least common multiple of 25. 8, 12, 16, 24 and 48? 27. 25, 40, 75, 80 and 120? 26. 16, 20, 24, 32 and 40? 28. 32, 45, 70, 64 and 90?

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29. What is the smallest nun,ber that will exactly contain 16, 24, and 30?

30. How long must a box be that no room may be lost in packing in it books 6 inches, 8 inches, or 12 inches long?

31. A lady desires to purchase a piece of cloth that can be cut without waste, into parts 4, 5, or 6 yards long. How many yards must the piece contain?

32. I have a certain number of pennies which I can arrange in either 4, 6, 8, 10, or 12 equal piles. What number of pennies have I, if it is the least number that admits of such arrangement?

33. How many bushels will the smallest bin contain that can be emptied by taking out either 7 bushels, 10 bushels, or 30 bushels at a time?

34. Four agents start from New York at the same time. The first makes his trip in 8 weeks, the second in 9 weeks, the third in 15 weeks, and the fourth in 20 weeks. How many weeks will pass by before they will again start out from New York together?

35. Three men walk around a circular island, the circumference of which is 360 miles. A walks 15 miles a day, B 18 miles a day, and C 24 miles a day. If they start together and walk in the same direction, how many days will elapse before they will be together again ?

36. Divide 5 x 15 X 80 X 56 X 81 by 10 X5 X 16 X 78.

37. If a man buys a lot whose sides measure respectively 48 feet, 60 feet, 96 feet and 108 feet, what will be the length of the longest boards which he can use to fence all the sides without cutting?

38. Find the greatest common divisor of 1744, 9564 and 8524.

39. What is the smallest number which can be divided by 250, 350, and 525 respectively, and leave a remainder of 25?

40. What is the greatest common divisor of 1728, 6912, and 8640?

41. A stock buyer wishes to invest the same amount of money in sheep at 83 each, hogs at $14 each, and cows at $21 each, as he does in beef cattle at $48 each. What is the smallest possible amount which he can invest in each ?

42. Jones Brothers & Co., of Cincinnati, O., received an order for a number of Lyman's Historical Charts. It was found that if the charts were packed in boxes containing either 24, 28, 32, or 36 charts each, there was a remainder of 9 each time, but if packed in boxes containing 25 each, there was no remainder. How many charts were ordered?

43. A dealer in real estate purchased 3 lots of land whose width on the street were respectively 152 rods, 288 rods, and 18+ rods. What is the width of the largest lots of equal size which can be formed from them?

41. Divide 3 X5 X 20 X 10 X 3X 13 by 26 X 9 X 3 X 4.

45. Find the greatest common divisor of 2219, 4501, and 5964.

46. Divide 5 X 8 X3 X 7 X 28 X 99 by 11 X 4X7X5 X 4.

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one-half

one-third

one-fourth

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fifth

140. 1. When a line is
divided into two equal parts,
what is each part called ?

22
2. When a line is divided
into three equal parts, what

13

12
is each part called ? What
are two of the parts called ?

14 14 14
3. When a line is divided

175 15 16 15 into four equals parts, what is each part called? What are two of the parts called ? What are three of the parts called ?

4. When any thing is divided into five equal parts, what is each part called? What are three parts called ? What are four parts called ?

5. When things are divided into 6, 7, 8, 9, 10, 15 equal parts respectively, what is each of the parts called? What are four of them called ?

6. How many halves are there in any thing? How many thirds? Fourths? Fifths? Sixths? Tenths ?

7. If 10 marbles are separated into 5 equal groups, what part of the marbles will be in each group?

8. How many are one-fifth of 10? Two-fifths ? Threefifths ? 9. How many are one-sixth of 12? Two-sixths?

1

DEFINITIONS.

141. A Fraction is one or more of the equal parts of a unit.

142. The Unit of a Fraction is the unit which is divided into equal parts.

A fraction in which the unit has been divided into any number of equal parts is called a Common Fraction.

A fraction in which the unit has been divided into tenths, hundredths, thousandths, etc., is called a Decimal Fraction.

143. A Fractional Unit is one of the equal parts into which a unit is divided.

144. Since a fraction is one or more of the equal parts of any thing, to express a fraction two numbers are necessary, one to express the number of equal parts into which the unit has -been divided, the other to express how many make the fraction. These numbers are written one above the other with a horizontal line between them.

145. The Denominator is the number which shows into how many equal parts the unit is divided.

It is written below the line.

Thus, in the fraction 1,7 is the denominator. It shows that the unit of the fraction has been divided into 7 equal parts.

146. The Numerator is the number which shows how many fractional units form the fraction.

It is written above the line.

Thus, in the fraction 4, 5 is the numerator and shows how many fractional units form the fraction.

147. The numerator and denominator are called the Terms of a Fraction.

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